Low-delay, Low-PAPR, High-rate Non-square Complex Orthogonal Designs

1 Lo w-delay , Lo w-P APR, High-ra te Non-square Comple x Orthogonal Designs Smarajit Das, Student Membe r , IEEE an d B. Sundar Rajan, Senior Member , IEEE Abstract —The maximal rate f or non-square Complex Or thog- onal Designs (CODs) with n transmit an tennas is 1 2 + 1 n if n is ev en and 1 2 + 1 n +1 if n is odd, which are close to 1 2 fo r large values of n. A class of ma ximal rate n on-square CODs ha ve been constructed by Liang (IEEE T rans. Info rm. Theory , 2003 ) and Lu et. al. (IEEE T rans. Inform. Theory , 2005) have shown that the decoding delay of th e codes giv en by Liang, can be reduced by 50% when n umber of transmit antenn as is a mul tiple of 4 . Adams et. al. (IEEE T rans. Inform. Theory , 2007) hav e shown that the designs of Liang are of minimal-delay for n eq ual to 1 and 3 modulo 4 and that of Lu et.al. are of minimal delay when n is a mul tiple of 4 . Howe ver , these minimal delays are large compar ed to the delays of the rate 1 / 2 non-sq uare CODs constructed by T arokh et al (IEEE T rans. Inform. Theory , 1999) from rate- 1 real orthogonal d esigns (RODs). In this paper , we construct a class of rate- 1 / 2 non-sq uare CODs f or any n with the decoding delay equal to 50% of that of the delay of the rate-1/2 codes gi ven by T arokh et al. This is achieved b y giving ﬁrst a general constructi on of rate-1 square Real Orthogonal Designs (RODs) wh ich includ es as sp ecial cases the well kn own constructions of Adams, Lax and Ph illips and Geramita and Pullman, and then making use of it to obtain th e desired rate- 1 2 non-square COD. For th e case of 9 transmit antennas, our rate- 1 2 COD is shown to be of minimal-delay . Th e proposed construction results in designs with zero entries which may hav e high Peak-to- A verag e Power Ratio (P APR) and i t is shown that by app ropriate postmultiplication, a design with no zero entries can be obtained with no change in the code parameters. I . I N T R O D U C T I O N A N D P R E L I M I N A R I E S There are se veral d eﬁnitions of Orthog onal Designs (ODs) in the literature [1 ], [7], [8] th e well known being as gi ven in [1]: Deﬁnition 1: A Complex Orthogon al Des ign (COD) G ( x 0 , x 1 , · · · , x k − 1 ) (in sho rt G ) for n tr ansmit a ntennas is deﬁned as a p × n matrix such that (i) the nonzero entries of G are the complex variables ± x 0 , ± x 1 , ..., ± x k − 1 and th eir conjuga tes and (ii) G H G = ( | x 0 | 2 + · · · + | x k − 1 | 2 ) I n where H stands for the comp lex conjugate tran spose and I n is the n × n identity matrix . The matrix G is also said to be a [ p, n, k ] COD and its rate in c omplex symbols per channel use is k p . When x 0 , · · · , x k − 1 are re al variables, the d esigns are called Real Orthogo nal Design ( R OD). This work was supported through grants to B.S. Rajan; partly by the IISc-DRDO program on Advanc ed Research in Mathema tical Engineerin g, and partly by the Council of S cient iﬁc & Industrial Research (CSIR, India ) Researc h Grant (22(0365)/04 /EMR-II). Pa rt of the materia l in t his paper w as presente d in IEE E 2008 Interna tional Symposium on Information Theory (ISIT -2008), July 6-11, T oronto, Cana da. Smarajit Das and B. Sundar Rajan are with the Department of Electrical Communicat ion Engineering, Indian Instit ute of Scienc e, Bangalore-560 012, India.emai l: { smaraj it,bsrajan } @ece.iisc.ernet.in. Space-time bloc k codes (STBCs) fro m CODs hav e been widely studied f or square designs, i.e., p = n, since they correspo nd to minimum d ecoding d elay co des for c o-located multiple-an tenna coheren t comm unication systems. Howev er , non-squ are designs naturally appear an d impo rtant in the following situatio ns. 1) In coheren t co- located MIM O systems, for a speciﬁed number o f tr ansmit a ntennas, n on-squa re designs can giv e m uch higher rate than the square designs [1]. 2) In non-co herent MIMO systems with non-differential detection, n on-squa re designs with p = 2 n lead to low decodin g comp lexity STBCs [2]. 3) Space-Time-Frequency co des can be viewed as n on- square designs [3]. 4) In d istributed sp ace-time cod ing for relay channe ls, rectangu lar designs natur ally appear [4]. 5) Rate 1 2 non-squ are CODs have been pr oposed for use in analog transmission with application to ch annel feed- back [5]. The rate of th e square CODs falls expo nentially with increase in the numb er of transmit antenn as. Th e f ollowing theorem relates the rate of a square OD, real/comp lex, with the number of tran smit antenn as. For an integer n = 2 a (2 b + 1) , where a = 4 c + d and a, b , c and d being integers with 0 ≤ d ≤ 3 , the Hurwitz-Radon n umber ρ ( n ) is deﬁne d as ρ ( n ) = 8 c + 2 d . Theor e m 1 ( [6], [7], [9]): The ma ximal rate of a sq uare R OD for n transmit a ntennas is g i ven by ρ ( n ) n where ρ ( n ) is the Hurwitz-Radon n umber of n , while that of a square COD is given by a +1 n where a is the exponen t of 2 in the prime factorization of n. Sev er al autho rs have constructed squar e CODs achieving maximal rate [6], [ 9]. In [6], th e following induction method is used to construct square CODs f or 2 a antennas, a = 2 , 3 , · · · , starting from G 1 =  x 0 − x ∗ 1 x 1 x ∗ 0  , G a =  G a − 1 − x ∗ a I 2 a − 1 x a I 2 a − 1 G H a − 1  , (1) where G a is a 2 a × 2 a complex m atrix. Note th at G a is a square COD in ( a + 1) complex variables x 0 , x 1 , x 2 , · · · , x a . It is clear fro m The orem 1 as well as the constructio n giv en by ( 1) that the square ODs, real and complex ar e not bandwidth efﬁcient and naturally on e is led to study non- square ODs in order to ob tain co des with higher rate. It is known that [7] ther e always exists a rate-1 R OD for any number of tr ansmit antenna s. In fact, the existence of rate - 1 [ p, n, p ] R OD is eq uiv alen t to that of a [ p, p, n ] square R OD. 2 For a rate-1 ROD, th e minimu m value o f decodin g delay p as a function of n , deno ted by ν ( n ) , is gi ven by ν ( n ) = 2 δ ( n ) where δ ( n ) = 8 > > > < > > > : 4 s if n = 8 s + 1 4 s + 1 if n = 8 s + 2 4 s + 2 if n = 8 s + 3 or 8 s + 4 4 s + 3 if n = 8 s + 5 , 8 s + 6 , 8 s + 7 or 8 s + 8 . (2) It is known [1] that the max imal rate of a non-sq uare COD is equ al to 1 2 + 1 2 a when the n umber o f transmit anten nas is 2 a − 1 or 2 a . Lia ng in [1] has given an explicit constru ction of non-squ are CODs achieving this max imal rate f or any numb er of an tennas. There is also ano ther constru ction of these codes giv en by Lu et al [10]. The fo rmer constru ction is algorithm ic in nature while the latter on e is based on patch-up of se veral matrices. The m inimum decod ing delay for the maximal r ate non-squ are CODs is, in g eneral, not k nown. The following theorem states what is kn own about the minimum delay of these code. For details, see [11]. Theor e m 2 ( [11]): A tigh t lower bou nd o n th e d ecoding delay of max imum ra te non-squa re CODs for n an tennas is  2 a a − 1  for n = 2 a − 1 or n = 2 a . Moreover , if n is co ngruen t to 0 , 1 or 3 mo dulo 4 , then this lo wer bound is achie vable. If n is co ngruen t to 2 m odulo 4 , the minimum deco ding de lay is upper bou nded b y 2  2 a a − 1  . As the rate of these maximal rate codes is close to 1 / 2 f or large number of antenn as, it is sufﬁcient to f ocus on rate 1 / 2 cod es when large number of ante nnas is unde r consideration. Then, a natural problem to study is constructio n of rate 1 / 2 non -square CODs with the decod ing d elay as small as p ossible. T arokh et al [ 7] have given a class of rate 1 / 2 codes ob tained fro m rate- 1 R ODs, which has lower delay s than those of maximal- rate codes constru cted in [1], [10] for n umber of transm it antenna s more than 5. For example, fo r f our transmit antenn as, the rate 1 / 2 code is G = 1 √ 2             x 0 − x 1 − x 2 − x 3 x 1 x 0 x 3 − x 2 x 2 − x 3 x 0 x 1 x 3 x 2 − x 1 x 0 x ∗ 0 − x ∗ 1 − x ∗ 2 − x ∗ 3 x ∗ 1 x ∗ 0 x ∗ 3 − x ∗ 2 x ∗ 2 − x ∗ 3 x ∗ 0 x ∗ 1 x ∗ 3 x ∗ 2 − x ∗ 1 x ∗ 0             . (3) Notice that, con trary to the deﬁnition in [ 7] of a COD, in this code, the variables appear in a colu mn more than once and each entr y o f all the co lumns of the design ma trix is scaled by 1 √ 2 in order to satisfy th e co ndition G H G = ( | x 0 | 2 + · · · + | x 3 | 2 ) I 4 of a COD. W e call such de signs scaled COD which is no t a COD in the conventional sense as in [1] (Deﬁnition 1). W e deﬁne the class of scaled CODs as fo llows : Deﬁnition 2: A λ -scaled complex o rthogonal design , for a positive integer λ, ( λ -scaled COD) G is a p × n matrix in k complex variables x 0 , x 1 , · · · , x k − 1 such th at a non-ze ro entry of the matrix is a variable or its co mplex con jugate, or the negative of these a nd all the entries of any subset of columns of the matrix is scaled by 1 √ λ satisfying the condition: G H G = ( | x 0 | 2 + · · · + | x k − 1 | 2 ) I n . The ma trix G is also said to be a [ p, n, k ] λ − scaled COD. Notice that a λ -scaled COD with with no colum n scaled by 1 √ λ is a COD. In column s with scaling b y 1 √ λ all th e variables appear exactly λ times. In this paper we co nsider only th e case λ = 2 an d call these codes simply scaled-COD. Contributions of this pa per: Th e contributions of th is paper may be summarized as follows: • For the rate 1 / 2 scaled CODs of [7], all the colum ns are scaled by the factor 1 √ 2 , which led to the r educed delay compar ed to the codes of Liang and a main result of this paper is that by having on ly a subset of the columns scaled b y 1 √ 2 further r eduction in d elay by 50 % is possible. W e u se following notations to refer to th e rate 1/2 CODs gi ven by T ar okh et al [7], the maximal rate codes giv en in Lu et al [10] an d th e codes of this paper . – L n is the maxim al rate COD f or n tran smit anten nas with the dec oding delay as speciﬁed in th e T heo- rem 2. – T J C n is the rate 1 / 2 scaled CODs f or n tran smit antennas constru cted by T aro kh et al [7]. – ( D R ) n is the rate 1 / 2 scaled CODs f or n transmit antennas constru cted in this p aper . Note that as n increases, the m aximal rate of L n ap- proach es 1 / 2 , thus two codes L n and T J C n can be compare d for large value of n, b ased on their delay s. It is not difﬁcult to see that the decoding delay of T J C n is less th an that of L n for large n . W e p rovide an explicit construction of rate 1 / 2 scaled CODs for any number of transmit antenna s such that the d ecoding delay of these codes is ν ( n ) when n is the numb er of transmit antennas, whereas th e d elay for the codes T J C n is 2 ν ( n ) . The T able I at th e top o f the next page shows that for large values of n, but for a marginal de crease in the rate with respect to L n , the co des of th is pap er are the b est codes. • For the case of 9 transmit ante nnas ou r ra te- 1 2 code is shown to be of m inimal-delay . • As a byprod uct of the above men tioned co nstruction, a general co nstruction o f squa re Real Orthog onal Designs (R ODs) is presented which inclu des a s special cases well known constru ctions of Ad ams, Lax an d Phillips [ 9] an d Geramita and Pullman [12]. • Even thoug h scalin g only a subset of c olumns allowed us to decr ease the delay , it is shown that such a scaling limits the ra te of the design strictly to 1 2 . In other word s, the maxima l rate of the scaled-CODs is 1 2 when scaling is present in atleast one column . • Zero entries in a design incr ease the Peak-to -A verage Power Ratio (P A PR) in the transmitted sign al and it is preferr ed not to have any ze ro entries in the design . This pro blem has been addre ssed for squ are design s in [13], [14]. All the k nown max imal rate non-squ are designs h av e zero entries. Ou r initial co nstruction of rate- 1 2 scaled CODs h av e zero en tries in the d esign matrix which will lead to hig her Peak-to-A verage Power Ratio (P APR) in con trast to the designs T J C n . Howe ver , we 3 T ABLE I T H E C O M PA R I S O N O F M A X I M U M R A T E AC H I E V I N G C O D E S A N D R AT E 1 / 2 C O D E S n 5 6 7 8 9 10 11 12 13 14 15 16 Decoding delay of ( DR ) n 8 8 8 8 16 32 64 64 128 128 128 128 Decoding delay of T J C n 16 16 16 16 32 64 128 128 256 256 256 256 Decoding delay of L n 15 30 56 56 210 420 792 792 3003 6006 11440 11440 Rate of ( D R ) n 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 Rate of T J C n 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 Rate of L n 2/3 2/3 5/8 5/8 3/5 3/5 7/12 7/12 4/7 4/7 9/16 9/16 show tha t by post-multiplication of approp riate matric es, our constru ction lea ds to designs with no zero entries without change in the parameter s of the d esign. The remainin g p art of the paper is organ ized as follows: In Section II, we present the main result of the p aper given by Theorem 5. Befo re th is, co nstruction of a n ew set of m aximal- rate square R ODs is given in Subsection II-A. In Subsectio n II-B, construction o f two new sets of rate-1 R ODs from the maximal-r ate square R ODs of Subsection II-A is pr esented and in Subsection II-C, construction of the low-delay ra te-1/2 scaled-CODs is achiev e d using rate-1 RODs of the p revious subsection. In Sub section II-D, it is shown that th e maxima l rate for scaled-CODs is 1 2 . For the special ca se of 9 transmit antennas, in Section III, it is shown that our constru ction is of m inimal d elay . In Section IV, we show tha t th e cod es discussed so far can be made to hav e no zer o entries in by appr opriate p reproce ssing without affecting the parameters of th e design. Conclu ding remarks co nstitute Section V. In Append ix B, it is shown tha t the well known constru ctions of square RODs by Adams-Lax- Phillips and Ger amita-Pullman are special cases of our constructio n. I I . A C O N S T R U C T I O N O F R A T E - 1 / 2 S C A L E D C O M P L E X O RT H O G O NA L D E S I G N S In this section , we construct a r ate-1/2 scaled CODs for any number of tr ansmit a ntennas with 50 % reduction in decodin g delay compare d to th e r ate-1/2 codes constructed by T aro kh e t al [7]. The co nstruction of th ese codes is don e in the f ollowing three steps: STEP 1: Construction of a new set o f square R ODs (Subsec- tion II-A). STEP 2: Con struction of two new sets of rate-1 R ODs fro m the square R ODs o f STEP 1 (Subsection II-B). STEP 3: Construction of low-delay rate-1/2 scaled CODs using rate-1 R ODs (Su bsection II-C). Before explainin g these steps, we ﬁrst build up some preliminar y results need ed to describe these steps. Let F 2 = { 0 , 1 } be th e ﬁnite ﬁeld w ith two elements with addition and multiplication denoted by b 1 ⊕ b 2 and b 1 b 2 for b 1 , b 2 ∈ F 2 . W e c onsider lo gical op erations also on the elements of this ﬁeld: b 1 + b 2 and ¯ b 1 represent respe cti vely the logical disjunctio n ( OR) of b 1 and b 2 and complem ent o r negation o f b 1 , i.e., b 1 + b 2 = b 1 ⊕ b 2 ⊕ b 1 b 2 , ¯ b 1 = 1 ⊕ b 1 . (4) For any ﬁnite subset B of the set of natural num bers N , let a ∈ N be the least integer such that b < 2 a for any b ∈ B . Often we identify each element of B with an element of F a 2 us- ing the following c orrespon dence: b ∈ B ↔ ( b a − 1 , · · · , b 0 ) ∈ F a 2 such that b = P a − 1 j =0 b j 2 j , b j ∈ F 2 . T he all zero vector a nd all one vector in F a 2 are denoted by 0 and 1 respectiv ely . For x ∈ B , x, x c and | x | repr esent respectively the 2 ′ s com plement of x in F a 2 , 1 ′ s complement of x in F a 2 and Hamming weig ht of x . In oth er words, x = 2 a − x and x c = 2 a − 1 − x. Let x = ( x a − 1 , · · · , x 0 ) and y = ( y a − 1 , · · · , y 0 ) . Then, x ⊕ y , x · y d enote the com ponen t-wise mod ulo-2 ad dition and compon ent-wise mu ltiplication (AND oper ation) of x and y respectively i.e., x ⊕ y = ( x a − 1 ⊕ y a − 1 , · · · , x 0 ⊕ y 0 ) and x · y = ( x a − 1 y a − 1 , · · · , x 0 y 0 ) . Let Z 2 a = { 0 , 1 , · · · , 2 a − 1 } . For a set K ⊂ Z 2 a , we deﬁne K = { x | x ∈ K } , K c = { x c | x ∈ K } , m ⊕ K = { m ⊕ a | a ∈ K } for any m ∈ Z 2 a and | K | denotes the number of elements in the set K . For any two sets A, B with B ⊂ A , the set A \ B , consists of those elements of A , which are no t in B . F o r two integers i, j , we use the notation i ≡ j , to indicate that i − j = 0 mo d 2 . For any matr ix of size n 1 × n 2 , th e rows and c olumns of the matrix are labeled by the elements o f { 0 , 1 , · · · , n 1 − 1 } and { 0 , 1 , · · · , n 2 − 1 } respectiv ely . If M is a p × n m atrix in k real variables x 0 , x 1 , x 2 , · · · , x k − 1 , such that each non -zero entry of the m atrix is x i or − x i for some i ∈ { 0 , 1 , · · · , k − 1 } , obviously , it is not n ecessary that M is a R OD. F o r exam ple,  x 0 x 1 x 1 x 0  is no t a R OD. I n the following ( Lemma 1), we derive a n ecessary an d sufﬁcient condition for th e matrix M to be a R OD. A submatrix M 2 of size 2 × 2 , constru cted by choosing any two rows and any two columns of M is called pr oper if • None of the entries of M 2 is zero and • It contains exactly two distinct v ariab les. Example 1: Co nsider the following m atrix in three real variables x 0 , x 1 and x 2 2 6 4 x 0 − x 1 − x 2 0 x 1 x 0 0 − x 2 x 2 0 x 0 x 1 0 x 2 − x 1 x 0 3 7 5 . (5) The sub-m atrix  x 1 − x 2 x 2 x 1  is pr ope r while  x 3 0 0 x 3  is not. If M ( i, j ) 6 = 0 , then we write | M ( i, j ) | = k whenever M ( i, j ) = x k or − x k ( in c ase of CODs, | M ( i, j ) | = k if M ( i, j ) ∈ {± x k , ± x ∗ k } ). 4 If M is a R OD and if M 2 is a 2 × 2 pr o per sub-matrix of M , containing two variables, say x l and x m , l, m p ositi ve integers, then M T 2 M 2 = ( x 2 l + x 2 m ) I 2 . In other words, M 2 is a R OD by itself in two variables. Th e f ollowing lemma gi ves a characteriza tion of R ODs in term of pro per 2 × 2 matrices. Lemma 1: L et M be a p × n matrix in k real variables x 0 , x 1 , x 2 , · · · , x k − 1 , such th at each non -zero entry of the matrix is x i or − x i for some i ∈ { 0 , 1 , · · · , k − 1 } . T hen the following two statements are equ i valent: 1) M is a R O D. 2) ( i) Each variable app ears exactly o nce alon g each column of M and atmo st once along each row of M , (ii) if for some i , j, j ′ , M ( i, j ) 6 = 0 and M ( i , j ′ ) 6 = 0 , then th ere exists i ′ such tha t | M ( i, j ) | = | M ( i ′ , j ′ ) | an d | M ( i, j ′ ) | = | M ( i ′ , j ) | , (iii) any proper 2 × 2 sub-m atrix of M is a R OD. A. STEP 1: Construction of a new class of squar e RODs Square RODs h a ve been construc ted b y sev eral autho rs, for exam ple, Adams et al [9] and Geramita e t al [12]. All these design s are co nstructed recu rsi vely and the basic blo cks of these designs are the R ODs o f order 1 , 2 , 4 and 8 . It is kn own that th ese designs ar e obtain ed by left (o r right) regular r epresentation s of the ﬁeld o f real number s, the ﬁeld of co mplex number s, the Qu aternion algeb ra and the Octonion algebra respectively . In th is subsection, we take a different approa ch towards the con struction of R OD s and that lead to a n e w class of R ODs constru ctable recursively of which the construction s of [9] an d [12] are special cases. If B t is a square real design of size [ t, t, k ] in k r eal v ariables x 0 , · · · , x k − 1 , then whenever B t ( i, j ) 6 = 0 , we write B t ( i, j ) = µ t ( i, j ) x λ t ( i,j ) , for some µ t ( i, j ) ∈ { 1 , − 1 } , and λ t ( i, j ) ∈ { 0 , 1 , · · · , k − 1 } for 0 ≤ i, j ≤ t − 1 . (6) B t is uniqu ely deter mined by µ t and λ t . The appr oach we take is identif ying a pair of functions µ t and λ t that deﬁnes a square R OD. T owards tha t end, for any t, iden tifying Z 2 a with F a 2 , w e have S ⊂ Z 2 a identiﬁed with a sub set of F a 2 . W e deﬁn e two maps wh ich are used fo r th e construction of our square R ODs as f ollows. Let γ t : Z ρ ( t ) → Z t (7) be an injective map deﬁned on Z ρ ( t ) with the image den oted by ˆ Z ρ ( t ) = γ t ( Z ρ ( t ) ) and ψ t : ˆ Z ρ ( t ) → Z t (8) be another injectiv e map deﬁn ed on ˆ Z ρ ( t ) . In th e fo llowing theorem, we d eﬁne the two maps µ t and λ t giv en in (6) which deﬁne the matrix B t , in terms of the two maps (7) and (8) and identify the condition s for the resulting B t to be a square R OD. Theor e m 3: Let t = 2 a and B t be a real d esign with B t ( i, j ) be non- zero if an d on ly if i ⊕ j ∈ ˆ Z ρ ( t ) . When B t ( i, j ) 6 = 0 , let µ t ( i, j ) = ( − 1) | i · ψ t ( i ⊕ j ) | , (9) λ t ( i, j ) = γ − 1 t ( i ⊕ j ) . (10) If | ( ψ t ( x ) ⊕ ψ t ( y )) · ( x ⊕ y ) | (11) is odd , for all x, y ∈ ˆ Z ρ ( t ) , x 6 = y , then, B t is a square ROD of size [ t, t, ρ ( t )] . Pr oof: W e use the Lemma 1 to prove th at B t is a R OD. First, for a ﬁxed j ∈ Z t , deﬁne A = { i ⊕ j | i ∈ Z t , i ⊕ j ∈ ˆ Z ρ ( t ) } . It is clear that A = ˆ Z ρ ( t ) . M oreover , as γ t is injectiv e, we have γ − 1 t ( i ⊕ j ) 6 = γ − 1 t ( i ′ ⊕ j ) wh enever i 6 = i ′ . Therefo re, each column o f the matrix B t contains all the variables x 0 , x 1 , · · · x ρ ( t ) − 1 and these variables a ppear exactly once. Similarly , it f ollows that the variables ap pear atmost once in any r ow o f B t . Secondly , assume that B t ( i, j ) 6 = 0 an d B t ( i, j ′ ) 6 = 0 , then we show tha t there exists i ′ such that | B t ( i, j ) | = | B t ( i ′ , j ′ ) | , | B t ( i, j ′ ) | = | B t ( i ′ , j ) | . Let i ′ = i ⊕ j ⊕ j ′ . W e have | B t ( i, j ) | = γ − 1 t ( i ⊕ j ) , | B t ( i ′ , j ′ ) | = γ − 1 t ( i ′ ⊕ j ′ ) = γ − 1 t ( i ⊕ j ) . Therefo re, | B t ( i, j ) | = | B t ( i ′ , j ′ ) | . Similarly , | B t ( i, j ′ ) | = | B t ( i ′ , j ) | . Third ly , we show th at any pr oper 2 × 2 sub-matr ix of B t is a R OD. It is enough to prove that µ t ( i, j ) · µ t ( i, j ′ ) · µ t ( i ′ , j ) · µ t ( i ′ , j ′ ) = − 1 , or equiv a lently , | i · ψ t ( i ⊕ j ) | + | i · ψ t ( i ⊕ j ′ ) | + | i ′ · ψ t ( i ′ ⊕ j ) | + | i ′ · ψ t ( i ′ ⊕ j ′ ) | (12) is an odd number . But i ′ = i ⊕ j ⊕ j ′ and | x ⊕ y | ≡ | x | + | y | where k ≡ l if k − l is a multiple of 2 . W e can write (1 2) as | ( i ⊕ i ′ ) · ( ψ t ( i ⊕ j ) ⊕ ψ t ( i ′ ⊕ j )) | = |  ( i ⊕ j ) ⊕ ( i ′ ⊕ j )) · ( ψ t ( i ⊕ j ) ⊕ ψ t ( i ′ ⊕ j )  | . which is an odd number as both i ⊕ j and i ′ ⊕ j ar e the elements of ˆ Z ρ ( t ) . Thus, it is eno ugh to co nstruct γ t and ψ t satisfying the proper ty stated in The orem 3, in order to construct a sq uare R OD. T his we a chieve by making u se of an other set of two maps φ 1 and φ 2 as follows. On Z 8 = { 0 , 1 , · · · , 7 } we de ﬁne a map φ 1 : Z 8 7→ Z 8 giv en by φ 1 =  0 1 2 3 4 5 6 7 0 1 2 3 4 7 5 6  . (13) For a ∈ { 0 , 1 , 2 , 3 } , we hav e Z 2 a ⊆ Z 8 . Deﬁne ˆ ψ 2 a : Z 2 a 7→ Z 2 a x 7→ φ 1 ( x ) (14) 5 where the 2 ’ s com plement of an elemen t x of F a 2 is perfo rmed in F a 2 . Note tha t th e m ap ˆ ψ 2 a is well deﬁned even though it is deﬁned in terms of φ 1 which is deﬁned on Z 8 . Lemma 2: L et x, y ∈ Z 2 a , a ∈ { 0 , 1 , 2 , 3 } , x 6 = y . Identify Z 2 a with F a 2 . Then | ( ˆ ψ 2 a ( x ) ⊕ ˆ ψ 2 a ( y )) · ( x ⊕ y ) | is an odd integer . Pr oof: W e prove it only f or a = 3 . For all other values of a , o ne can p rove it similarly . Write ˆ x = ˆ ψ 8 ( x ) . L et the radix-2 repr esentation o f x ∈ Z 8 be ( x 2 , x 1 , x 0 ) an d th at of ˆ x be ( ˆ x 2 , ˆ x 1 , ˆ x 0 ) wh ere each x i or ˆ x i takes value fr om th e set F 2 = { 0 , 1 } . The f ollowing table describes the map x → ˆ x for x = 0 , · · · , 7 . x x 2 x 1 x 0 ˆ x 2 ˆ x 1 ˆ x 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 2 0 1 0 1 1 0 3 0 1 1 1 0 1 4 1 0 0 1 0 0 5 1 0 1 0 0 1 6 1 1 0 0 1 1 7 1 1 1 0 1 0 Using this table, and (4) we express ˆ x i in term of x 2 , x 1 and x 0 for i = 0 , 1 and 2 as follows. ˆ x 0 = ¯ x 2 x 0 + x 2 ( x 1 ⊕ x 0 ) = x 0 ⊕ x 1 x 2 , ˆ x 1 = x 2 x 1 + ¯ x 2 ( x 1 ⊕ x 0 ) = x 0 ⊕ x 1 ⊕ x 2 x 0 , ˆ x 2 = ¯ x 2 x 1 + ¯ x 1 ( x 2 ⊕ x 0 ) = x 0 ⊕ x 1 ⊕ x 2 ⊕ x 0 x 1 . Hence ˆ x i = i X j =0 x j ⊕ Y 0 ≤ j ≤ 2 j 6 = i x j , i = 0 , 1 an d 2 . Let x, y ∈ Z 8 , x 6 = y . Now | ( ˆ x ⊕ ˆ y ) · ( x ⊕ y ) | is odd if an d only if 2 X i =0 ( x i ⊕ y i )( ˆ x i ⊕ ˆ y i ) = 1 . But P 2 i =0 ( x i ⊕ y i )( ˆ x i ⊕ ˆ y i ) = 1 ⊕ Q 2 i =0 (1 ⊕ x i ⊕ y i ) . Now 1 ⊕ Q 2 i =0 (1 ⊕ x i ⊕ y i ) is equal to 0 if and only if Q 2 j =0 (1 ⊕ x i ⊕ y i ) = 1 i.e. , ( x i ⊕ y i ) = 0 fo r all i , which implies that x = y . As x 6 = y , we hav e P 2 i =0 ( x i ⊕ y i )( ˆ x i ⊕ ˆ y i ) = 1 . The second map φ 2 is deﬁned on the set F given by F = n x ∈ F 4 2    | x | = 1 or 3 o = { 1 , 2 , 4 , 7 , 8 , 1 1 , 13 , 1 4 } (15) as injectiv e map φ 2 : F 7→ Z 16 giv en by φ 2 =  1 2 4 7 8 11 13 14 1 2 4 6 8 15 10 12  . (16 ) Lemma 3: L et F be the set given in (1 5) and x, y ∈ F, x 6 = y . Th en (i) | φ 2 ( x ) · x | is o dd for all x 6 = 0 . (ii) | φ 2 ( x ) · y | + | φ 2 ( y ) · x | is odd f or all x 6 = y , x 6 = 0 , y 6 = 0 . Pr oof: Ther e are only ﬁnitely many possibilities for x and y and it can be checked that both the statements (i) and (ii) hold for all possible cases. Now , we deﬁn e th e maps γ t and ψ t in terms of the maps φ 1 and φ 2 as fo llows: T he map γ t deﬁned over Z ρ ( t ) is given by γ t ( i ) = ( i if 0 ≤ i ≤ 7 2 4 l − 1 · ˆ γ ( m ) if i ≥ 8 , i = 8 l + m , 0 ≤ m ≤ 7 (17) where ˆ γ =  0 1 2 3 4 5 6 7 1 2 4 7 8 11 13 14  . (18) For an element x ∈ ˆ Z ρ ( t ) , either x ∈ Z 8 or x = 2 4 y − 1 z f or some y ∈ N \ { 0 } and z ∈ F . Let φ b e the map deﬁn ed on the set ˆ Z ρ ( t ) giv en by φ ( x ) = ( φ 1 ( x ) if x ∈ Z 8 2 4 y − 1 · φ 2 ( z ) if x = 2 4 y − 1 z , z ∈ F . (19) The map ψ t is deﬁned by ψ t ( x ) = φ ( x ) in F a 2 ∀ x ∈ ˆ Z ρ ( t ) . (20) The following th eorem shows that the m aps γ t and ψ t deﬁned by (17) and (20) satisfy the con ditions of Theo rem 3 and hence deﬁne a R OD. Theor e m 4: Id entify ˆ Z ρ ( t ) with a subset of F a 2 , t = 2 a . Then | ( ψ t ( x ) ⊕ ψ t ( y )) · ( x ⊕ y ) | is odd for all x, y ∈ ˆ Z ρ ( t ) , x 6 = y , and hence f rom Theorem 3 th e matrix B t deﬁned by γ t and ψ t by (17) and (20) is a square R OD. Pr oof: For t = 1 , 2 , 4 and 8 , the statement hold s by Lemma 2. Hence we assum e th at t ≥ 16 . As ψ t (0) = 0 , it is enough to prove that (i) | ψ t ( y ) · y | is odd for a ll y 6 = 0 . (ii) | ψ t ( x ) · y | + | ψ t ( y ) · x | is odd for all x 6 = y , x 6 = 0 , y 6 = 0 . T o prove (i) , let z = ψ t ( y ) · y . If y ∈ E , we have | ψ t ( y ) · y | = | ψ 8 ( y ) · y | which is an odd num ber b y Lem ma 2. On the o ther hand, if y = 2 4 l − 1 m , l ≥ 0 , m ∈ F , then | z | = | 2 4 l − 1 φ 2 ( m ) · 2 4 l − 1 m | wh ere the 2 ′ s co mplement of an element is per formed in F a 2 . W e h av e | z | = | φ 2 ( m ) · m | wh ere the 2 ′ s complemen t of φ 2 ( m ) is perf ormed in F 4 2 . Hence | z | is odd by Lemma 3. In order to prove the part (ii), we have following three cases: (i) 1 ≤ x ≤ 7 & 1 ≤ y ≤ 7 , (ii) 1 ≤ y ≤ 7 & x = 2 4 α − 1 β for some β ∈ F , α ≥ 1 , (iii) x = 2 4 ˆ α − 1 ˆ β & y = 2 4 α − 1 β for some β , ˆ β ∈ F, α, ˆ α ≥ 1 . In all the three cases, we have x 6 = y . By Lemma 2, (i) is true. For the secon d case, let z = ψ t ( x ) · y ⊕ ψ t ( y ) · x . W e hav e z = ( 2 4 α − 1 φ 2 ( β ) · y ) ⊕ ((2 4 α − 1 β ) · φ 1 ( y )) . As 2 4 α − 1 φ 2 ( β ) · y = 0 (the a ll zero vector in F a 2 ) fo r α ≥ 1 , we have z = (2 4 α − 1 β ) · φ 1 ( y ) . But | β | is od d for all β ∈ F , hence | z | is an o dd number . For (iii), let z = ψ t ( x ) · y ⊕ ψ t ( y ) · x . W e have z = 2 4 α − 1 φ 2 ( β ) · 2 4 ˆ α − 1 ˆ β ⊕ 2 4 α − 1 β · 2 4 ˆ α − 1 φ 2 ( ˆ β ) . 6 If ˆ α > α , we h av e 2 4 α − 1 β · 2 4 ˆ α − 1 φ 2 ( ˆ β ) = 0 and 2 4 α − 1 φ 2 ( β ) · 2 4 ˆ α − 1 ˆ β = ˆ β . Th us | z | is an odd num ber by Lemma 3. If α = ˆ α , it fo llows that | z | = | φ 2 ( β ) · ˆ β | + | β · φ 2 ( ˆ β ) | which is an odd number by Lemma 3. From Th eorem 3 it fo llows th at the matrix B t deﬁned b y γ t and ψ t by (17) and (20) is a square R OD. The square RODs o f T heorem 4 will be deno ted by R t throug hout. The R O D R 16 of size [16 , 16 , 9 ] is given by (2 1) at the top of the next page. As anoth er example th e ROD R 32 of size [32 , 32 , 10] is given b y (2 2) in the following page. In Append ix A, it is shown that th e R ODs R t can be c onstructed recursively . One can deﬁn e the functions γ t and ψ t different fro m the one g i ven above and can have a square ROD different from R t . In Appendix B, we provide th ree different pair s of such function s and th ese are shown to give the we ll known Ad ams- Lax-Phillips’ con struction from Octon ions a nd Quatern ions and Geramita and Pullman’ s constructio n o f square R ODs. B. STEP 2 : Construction of new sets of rate-1 RODs T ra nsition from a square R OD to rate-1 R OD is illustrated in [7] u sing column vector rep resentation of a R OD. In a similar way , we construct a rate-1 R OD W n of size [ ν ( n ) , n, ν ( n )] fo r n transmit an tennas from a ROD o f size [ ν ( n ) , ν ( n ) , n ] wh ere n is any n on-zero positive integer, no t n ecessarily power o f 2 . Any square ROD of o rder ν ( n ) obtained via a suitable pair of mapp ing γ ν ( n ) and ψ ν ( n ) satisfying the cond itions of Theorem 3 ( for in stance, R ν ( n ) obtained in th e p revious subsection or A ν ( n ) , ˆ A ν ( n ) and G ν ( n ) obtained in Ap pendix B) can be used for this purpose. W e ref er to any such design by B ν ( n ) consisting of n real variables z 0 , z 1 , · · · , z n − 1 . Let y 0 , y 1 , · · · , y ν ( n ) − 1 be ν ( n ) real variables which con - stitute the matr ix W n . T he matrix W n is obtained as follows: Make W n ( i, j ) = 0 if the i -th row of B ν ( n ) does not contain z j . Otherwise, W n ( i, j ) = y k or − y k if B ν ( n ) ( i, k ) = z j or − z j respectively . The con struction of the m atrix W n ensures that it is a rate-1 R OD. Using (6),( 9) and Theor em 4, we have W n ( i, j ) = s ( i, j ) y f ( i,j ) , (23) where s ( i, j ) = ( − 1) | i · ψ ν ( n ) ( γ ν ( n ) ( j )) | (24) f ( i, j ) = i ⊕ γ ν ( n ) ( j ) for 0 ≤ i ≤ ν ( n ) − 1 , 0 ≤ j ≤ n − 1 . Let ˆ W n be a matrix which is similar to the m atrix W n , g iv en by ˆ W n ( i, j ) = ˆ s ( i, j ) y f ( i,j ) , (25) where ˆ s ( i, j ) = ( − 1 ) | ( i ⊕ γ ν ( n ) ( j )) · ψ ν ( n ) ( γ ν ( n ) ( j )) | (26) f ( i, j ) = i ⊕ γ ν ( n ) ( j ) . ˆ W n is also a rate-1 R OD. Both W n and ˆ W n are the new sets of rate-1 R ODs th at are u sed in the following sub section to construct our codes. As examples, the RODs W 9 and ˆ W 9 of size [16 , 9 , 1 6] obtained using R 16 are giv en by (27) and (28) respectively and the R ODs W 10 and ˆ W 10 of size [3 2 , 10 , 32 ] obtained usin g R 32 are given by (29) and (30) resp ecti vely . W 9 = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 y 0 y 1 y 2 y 3 y 4 y 5 y 6 y 7 y 8 y 1 − y 0 y 3 − y 2 y 5 − y 4 − y 7 y 6 y 9 y 2 − y 3 − y 0 y 1 y 6 y 7 − y 4 − y 5 y 10 y 3 y 2 − y 1 − y 0 y 7 − y 6 y 5 − y 4 y 11 y 4 − y 5 − y 6 − y 7 − y 0 y 1 y 2 y 3 y 12 y 5 y 4 − y 7 y 6 − y 1 − y 0 − y 3 y 2 y 13 y 6 y 7 y 4 − y 5 − y 2 y 3 − y 0 − y 1 y 14 y 7 − y 6 y 5 y 4 − y 3 − y 2 y 1 − y 0 y 15 y 8 − y 9 − y 10 − y 11 − y 12 − y 13 − y 14 − y 15 − y 0 y 9 y 8 − y 11 y 10 − y 13 y 12 y 15 − y 14 − y 1 y 10 y 11 y 8 − y 9 − y 14 − y 15 y 12 y 13 − y 2 y 11 − y 10 y 9 y 8 − y 15 y 14 − y 13 y 12 − y 3 y 12 y 13 y 14 y 15 y 8 − y 9 − y 10 − y 11 − y 4 y 13 − y 12 y 15 − y 14 y 9 y 8 y 11 − y 10 − y 5 y 14 − y 15 − y 12 y 13 y 10 − y 11 y 8 y 9 − y 6 y 15 y 14 − y 13 − y 12 y 11 y 10 − y 9 y 8 − y 7 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 (27) ˆ W 9 = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 y 0 − y 1 − y 2 y 3 − y 4 y 5 y 6 − y 7 − y 8 y 1 y 0 − y 3 − y 2 − y 5 − y 4 − y 7 − y 6 − y 9 y 2 y 3 y 0 y 1 − y 6 y 7 − y 4 y 5 − y 10 y 3 − y 2 y 1 − y 0 − y 7 − y 6 y 5 y 4 − y 11 y 4 y 5 y 6 − y 7 y 0 y 1 y 2 − y 3 − y 12 y 5 − y 4 y 7 y 6 y 1 − y 0 − y 3 − y 2 − y 13 y 6 − y 7 − y 4 − y 5 y 2 y 3 − y 0 y 1 − y 14 y 7 y 6 − y 5 y 4 y 3 − y 2 y 1 y 0 − y 15 y 8 y 9 y 10 − y 11 y 12 − y 13 − y 14 y 15 y 0 y 9 − y 8 y 11 y 10 y 13 y 12 y 15 y 14 y 1 y 10 − y 11 − y 8 − y 9 y 14 − y 15 y 12 − y 13 y 2 y 11 y 10 − y 9 y 8 y 15 y 14 − y 13 − y 12 y 3 y 12 − y 13 − y 14 y 15 − y 8 − y 9 − y 10 y 11 y 4 y 13 y 12 − y 15 − y 14 − y 9 y 8 y 11 y 10 y 5 y 14 y 15 y 12 y 13 − y 10 − y 11 y 8 − y 9 y 6 y 15 − y 14 y 13 − y 12 − y 11 y 10 − y 9 − y 8 y 7 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 (28) W 10 = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 y 0 y 1 y 2 y 3 y 4 y 5 y 6 y 7 y 8 y 16 y 1 − y 0 y 3 − y 2 y 5 − y 4 − y 7 y 6 y 9 y 17 y 2 − y 3 − y 0 y 1 y 6 y 7 − y 4 − y 5 y 10 y 18 y 3 y 2 − y 1 − y 0 y 7 − y 6 y 5 − y 4 y 11 y 19 y 4 − y 5 − y 6 − y 7 − y 0 y 1 y 2 y 3 y 12 y 20 y 5 y 4 − y 7 y 6 − y 1 − y 0 − y 3 y 2 y 13 y 21 y 6 y 7 y 4 − y 5 − y 2 y 3 − y 0 − y 1 y 14 y 22 y 7 − y 6 y 5 y 4 − y 3 − y 2 y 1 − y 0 y 15 y 23 y 8 − y 9 − y 10 − y 11 − y 12 − y 13 − y 14 − y 15 − y 0 y 24 y 9 y 8 − y 11 y 10 − y 13 y 12 y 15 − y 14 − y 1 y 25 y 10 y 11 y 8 − y 9 − y 14 − y 15 y 12 y 13 − y 2 y 26 y 11 − y 10 y 9 y 8 − y 15 y 14 − y 13 y 12 − y 3 y 27 y 12 y 13 y 14 y 15 y 8 − y 9 − y 10 − y 11 − y 4 y 28 y 13 − y 12 y 15 − y 14 y 9 y 8 y 11 − y 10 − y 5 y 29 y 14 − y 15 − y 12 y 13 y 10 − y 11 y 8 y 9 − y 6 y 30 y 15 y 14 − y 13 − y 12 y 11 y 10 − y 9 y 8 − y 7 y 31 y 16 − y 17 − y 18 − y 19 − y 20 − y 21 − y 22 − y 23 − y 24 − y 0 y 17 y 16 − y 19 y 18 − y 21 y 20 y 23 − y 22 − y 25 − y 1 y 18 y 19 y 16 − y 17 − y 22 − y 23 y 20 y 21 − y 26 − y 2 y 19 − y 18 y 17 y 16 − y 23 y 22 − y 21 y 20 − y 27 − y 3 y 20 y 21 y 22 y 23 y 16 − y 17 − y 18 − y 19 − y 28 − y 4 y 21 − y 20 y 23 − y 22 y 17 y 16 y 19 − y 18 − y 29 − y 5 y 22 − y 23 − y 20 y 21 y 18 − y 19 y 16 y 17 − y 30 − y 6 y 23 y 22 − y 21 − y 20 y 19 y 18 − y 17 y 16 − y 31 − y 7 y 24 y 25 y 26 y 27 y 28 y 29 y 30 y 31 y 16 − y 8 y 25 − y 24 y 27 − y 26 y 29 − y 28 − y 31 y 30 y 17 − y 9 y 26 − y 27 − y 24 y 25 y 30 y 31 − y 28 − y 29 y 18 − y 10 y 27 y 26 − y 25 − y 24 y 31 − y 30 y 29 − y 28 y 19 − y 11 y 28 − y 29 − y 30 − y 31 − y 24 y 25 y 26 y 27 y 20 − y 12 y 29 y 28 − y 31 y 30 − y 25 − y 24 − y 27 y 26 y 21 − y 13 y 30 y 31 y 28 − y 29 − y 26 y 27 − y 24 − y 25 y 22 − y 14 y 31 − y 30 y 29 y 28 − y 27 − y 26 y 25 − y 24 y 23 − y 15 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 (29) 7 R 16 =                      x 0 x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 0 0 0 0 0 0 0 − x 1 x 0 − x 3 x 2 − x 5 x 4 x 7 − x 6 0 x 8 0 0 0 0 0 0 − x 2 x 3 x 0 − x 1 − x 6 − x 7 x 4 x 5 0 0 x 8 0 0 0 0 0 − x 3 − x 2 x 1 x 0 − x 7 x 6 − x 5 x 4 0 0 0 x 8 0 0 0 0 − x 4 x 5 x 6 x 7 x 0 − x 1 − x 2 − x 3 0 0 0 0 x 8 0 0 0 − x 5 − x 4 x 7 − x 6 x 1 x 0 x 3 − x 2 0 0 0 0 0 x 8 0 0 − x 6 − x 7 − x 4 x 5 x 2 − x 3 x 0 x 1 0 0 0 0 0 0 x 8 0 − x 7 x 6 − x 5 − x 4 x 3 x 2 − x 1 x 0 0 0 0 0 0 0 0 x 8 − x 8 0 0 0 0 0 0 0 x 0 − x 1 − x 2 − x 3 − x 4 − x 5 − x 6 − x 7 0 − x 8 0 0 0 0 0 0 x 1 x 0 x 3 − x 2 x 5 − x 4 − x 7 x 6 0 0 − x 8 0 0 0 0 0 x 2 − x 3 x 0 x 1 x 6 x 7 − x 4 − x 5 0 0 0 − x 8 0 0 0 0 x 3 x 2 − x 1 x 0 x 7 − x 6 x 5 − x 4 0 0 0 0 − x 8 0 0 0 x 4 − x 5 − x 6 − x 7 x 0 x 1 x 2 x 3 0 0 0 0 0 − x 8 0 0 x 5 x 4 − x 7 x 6 − x 1 x 0 − x 3 x 2 0 0 0 0 0 0 − x 8 0 x 6 x 7 x 4 − x 5 − x 2 x 3 x 0 − x 1 0 0 0 0 0 0 0 − x 8 x 7 − x 6 x 5 x 4 − x 3 − x 2 x 1 x 0                      (21) R 32 = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 x 0 x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 0 0 0 0 0 0 0 x 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 − x 1 x 0 − x 3 x 2 − x 5 x 4 x 7 − x 6 0 x 8 0 0 0 0 0 0 0 x 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 − x 2 x 3 x 0 − x 1 − x 6 − x 7 x 4 x 5 0 0 x 8 0 0 0 0 0 0 0 x 9 0 0 0 0 0 0 0 0 0 0 0 0 0 − x 3 − x 2 x 1 x 0 − x 7 x 6 − x 5 x 4 0 0 0 x 8 0 0 0 0 0 0 0 x 9 0 0 0 0 0 0 0 0 0 0 0 0 − x 4 x 5 x 6 x 7 x 0 − x 1 − x 2 − x 3 0 0 0 0 x 8 0 0 0 0 0 0 0 x 9 0 0 0 0 0 0 0 0 0 0 0 − x 5 − x 4 x 7 − x 6 x 1 x 0 x 3 − x 2 0 0 0 0 0 x 8 0 0 0 0 0 0 0 x 9 0 0 0 0 0 0 0 0 0 0 − x 6 − x 7 − x 4 x 5 x 2 − x 3 x 0 x 1 0 0 0 0 0 0 x 8 0 0 0 0 0 0 0 x 9 0 0 0 0 0 0 0 0 0 − x 7 x 6 − x 5 − x 4 x 3 x 2 − x 1 x 0 0 0 0 0 0 0 0 x 8 0 0 0 0 0 0 0 x 9 0 0 0 0 0 0 0 0 − x 8 0 0 0 0 0 0 0 x 0 − x 1 − x 2 − x 3 − x 4 − x 5 − x 6 − x 7 0 0 0 0 0 0 0 0 x 9 0 0 0 0 0 0 0 0 − x 8 0 0 0 0 0 0 x 1 x 0 x 3 − x 2 x 5 − x 4 − x 7 x 6 0 0 0 0 0 0 0 0 0 x 9 0 0 0 0 0 0 0 0 − x 8 0 0 0 0 0 x 2 − x 3 x 0 x 1 x 6 x 7 − x 4 − x 5 0 0 0 0 0 0 0 0 0 0 x 9 0 0 0 0 0 0 0 0 − x 8 0 0 0 0 x 3 x 2 − x 1 x 0 x 7 − x 6 x 5 − x 4 0 0 0 0 0 0 0 0 0 0 0 x 9 0 0 0 0 0 0 0 0 − x 8 0 0 0 x 4 − x 5 − x 6 − x 7 x 0 x 1 x 2 x 3 0 0 0 0 0 0 0 0 0 0 0 0 x 9 0 0 0 0 0 0 0 0 − x 8 0 0 x 5 x 4 − x 7 x 6 − x 1 x 0 − x 3 x 2 0 0 0 0 0 0 0 0 0 0 0 0 0 x 9 0 0 0 0 0 0 0 0 − x 8 0 x 6 x 7 x 4 − x 5 − x 2 x 3 x 0 − x 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x 9 0 0 0 0 0 0 0 0 − x 8 x 7 − x 6 x 5 x 4 − x 3 − x 2 x 1 x 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x 9 − x 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x 0 − x 1 − x 2 − x 3 − x 4 − x 5 − x 6 − x 7 − x 8 0 0 0 0 0 0 0 0 − x 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x 1 x 0 x 3 − x 2 x 5 − x 4 − x 7 x 6 0 − x 8 0 0 0 0 0 0 0 0 − x 9 0 0 0 0 0 0 0 0 0 0 0 0 0 x 2 − x 3 x 0 x 1 x 6 x 7 − x 4 − x 5 0 0 − x 8 0 0 0 0 0 0 0 0 − x 9 0 0 0 0 0 0 0 0 0 0 0 0 x 3 x 2 − x 1 x 0 x 7 − x 6 x 5 − x 4 0 0 0 − x 8 0 0 0 0 0 0 0 0 − x 9 0 0 0 0 0 0 0 0 0 0 0 x 4 − x 5 − x 6 − x 7 x 0 x 1 x 2 x 3 0 0 0 0 − x 8 0 0 0 0 0 0 0 0 − x 9 0 0 0 0 0 0 0 0 0 0 x 5 x 4 − x 7 x 6 − x 1 x 0 − x 3 x 2 0 0 0 0 0 − x 8 0 0 0 0 0 0 0 0 − x 9 0 0 0 0 0 0 0 0 0 x 6 x 7 x 4 − x 5 − x 2 x 3 x 0 − x 1 0 0 0 0 0 0 − x 8 0 0 0 0 0 0 0 0 − x 9 0 0 0 0 0 0 0 0 x 7 − x 6 x 5 x 4 − x 3 − x 2 x 1 x 0 0 0 0 0 0 0 0 − x 8 0 0 0 0 0 0 0 0 − x 9 0 0 0 0 0 0 0 x 8 0 0 0 0 0 0 0 x 0 x 1 x 2 x 3 x 4 x 5 x 6 x 7 0 0 0 0 0 0 0 0 0 − x 9 0 0 0 0 0 0 0 x 8 0 0 0 0 0 0 − x 1 x 0 − x 3 x 2 − x 5 x 4 x 7 − x 6 0 0 0 0 0 0 0 0 0 0 − x 9 0 0 0 0 0 0 0 x 8 0 0 0 0 0 − x 2 x 3 x 0 − x 1 − x 6 − x 7 x 4 x 5 0 0 0 0 0 0 0 0 0 0 0 − x 9 0 0 0 0 0 0 0 x 8 0 0 0 0 − x 3 − x 2 x 1 x 0 − x 7 x 6 − x 5 x 4 0 0 0 0 0 0 0 0 0 0 0 0 − x 9 0 0 0 0 0 0 0 x 8 0 0 0 − x 4 x 5 x 6 x 7 x 0 − x 1 − x 2 − x 3 0 0 0 0 0 0 0 0 0 0 0 0 0 − x 9 0 0 0 0 0 0 0 x 8 0 0 − x 5 − x 4 x 7 − x 6 x 1 x 0 x 3 − x 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 − x 9 0 0 0 0 0 0 0 x 8 0 − x 6 − x 7 − x 4 x 5 x 2 − x 3 x 0 x 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 − x 9 0 0 0 0 0 0 0 x 8 − x 7 x 6 − x 5 − x 4 x 3 x 2 − x 1 x 0 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 (22) ˆ W 10 = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 y 0 − y 1 − y 2 y 3 − y 4 y 5 y 6 − y 7 − y 8 − y 16 y 1 y 0 − y 3 − y 2 − y 5 − y 4 − y 7 − y 6 − y 9 − y 17 y 2 y 3 y 0 y 1 − y 6 y 7 − y 4 y 5 − y 10 − y 18 y 3 − y 2 y 1 − y 0 − y 7 − y 6 y 5 y 4 − y 11 − y 19 y 4 y 5 y 6 − y 7 y 0 y 1 y 2 − y 3 − y 12 − y 20 y 5 − y 4 y 7 y 6 y 1 − y 0 − y 3 − y 2 − y 13 − y 21 y 6 − y 7 − y 4 − y 5 y 2 y 3 − y 0 y 1 − y 14 − y 22 y 7 y 6 − y 5 y 4 y 3 − y 2 y 1 y 0 − y 15 − y 23 y 8 y 9 y 10 − y 11 y 12 − y 13 − y 14 y 15 y 0 − y 24 y 9 − y 8 y 11 y 10 y 13 y 12 y 15 y 14 y 1 − y 25 y 10 − y 11 − y 8 − y 9 y 14 − y 15 y 12 − y 13 y 2 − y 26 y 11 y 10 − y 9 y 8 y 15 y 14 − y 13 − y 12 y 3 − y 27 y 12 − y 13 − y 14 y 15 − y 8 − y 9 − y 10 y 11 y 4 − y 28 y 13 y 12 − y 15 − y 14 − y 9 y 8 y 11 y 10 y 5 − y 29 y 14 y 15 y 12 y 13 − y 10 − y 11 y 8 − y 9 y 6 − y 30 y 15 − y 14 y 13 − y 12 − y 11 y 10 − y 9 − y 8 y 7 − y 31 y 16 y 17 y 18 − y 19 y 20 − y 21 − y 22 y 23 y 24 y 0 y 17 − y 16 y 19 y 18 y 21 y 20 y 23 y 22 y 25 y 1 y 18 − y 19 − y 16 − y 17 y 22 − y 23 y 20 − y 21 y 26 y 2 y 19 y 18 − y 17 y 16 y 23 y 22 − y 21 − y 20 y 27 y 3 y 20 − y 21 − y 22 y 23 − y 16 − y 17 − y 18 y 19 y 28 y 4 y 21 y 20 − y 23 − y 22 − y 17 y 16 y 19 y 18 y 29 y 5 y 22 y 23 y 20 y 21 − y 18 − y 19 y 16 − y 17 y 30 y 6 y 23 − y 22 y 21 − y 20 − y 19 y 18 − y 17 − y 16 y 31 y 7 y 24 − y 25 − y 26 y 27 − y 28 y 29 y 30 − y 31 − y 16 y 8 y 25 y 24 − y 27 − y 26 − y 29 − y 28 − y 31 − y 30 − y 17 y 9 y 26 y 27 y 24 y 25 − y 30 y 31 − y 28 y 29 − y 18 y 10 y 27 − y 26 y 25 − y 24 − y 31 − y 30 y 29 y 28 − y 19 y 11 y 28 y 29 y 30 − y 31 y 24 y 25 y 26 − y 27 − y 20 y 12 y 29 − y 28 y 31 y 30 y 25 − y 24 − y 27 − y 26 − y 21 y 13 y 30 − y 31 − y 28 − y 29 y 26 y 27 − y 24 y 25 − y 22 y 14 y 31 y 30 − y 29 y 28 y 27 − y 26 y 25 y 24 − y 23 y 15 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 (30) C. STEP 3 : Constructio n of low-d elay rate-1/2 Scaled CODs In this subsection, we c onstruct a rate-1 /2 scaled COD with the help of rate-1 R ODs W n and ˆ W n constructed in the previous sub section. Let x 0 , x 1 , · · · b e com plex variables. The 8 × 8 , r ate- 1 2 CODs A ( x 0 , x 1 , x 2 , x 3 ) and B ( x 4 , x 5 , x 6 , x 7 ) and the 8 × 1 column vecto r C ( x 0 , x 1 , x 2 , x 3 ) in four variables, shown be- low , are th e basic ing redients for o ur construc tion o f rate- 1 2 scaled CODs. A ( x 0 , x 1 , x 1 , x 3 ) = 2 6 6 6 6 6 6 6 6 4 x 0 − x ∗ 1 − x ∗ 2 0 − x ∗ 3 0 0 0 x 1 x ∗ 0 0 − x ∗ 2 0 − x ∗ 3 0 0 x 2 0 x ∗ 0 x ∗ 1 0 0 − x ∗ 3 0 0 x 2 − x 1 x 0 0 0 0 − x ∗ 3 x 3 0 0 0 x ∗ 0 x ∗ 1 x ∗ 2 0 0 x 3 0 0 − x 1 x 0 0 x ∗ 2 0 0 x 3 0 − x 2 0 x 0 − x ∗ 1 0 0 0 x 3 0 − x 2 x 1 x ∗ 0 3 7 7 7 7 7 7 7 7 5 (31) B ( x 4 , x 5 , x 6 , x 7 ) = 2 6 6 6 6 6 6 6 6 4 x 4 − x ∗ 5 − x ∗ 6 − x ∗ 7 0 0 0 0 x 5 x ∗ 4 0 0 − x ∗ 6 − x ∗ 7 0 0 x 6 0 x ∗ 4 0 x ∗ 5 0 − x ∗ 7 0 0 x 6 − x 5 0 x 4 0 0 − x ∗ 7 x 7 0 0 x ∗ 4 0 x ∗ 5 x ∗ 6 0 0 x 7 0 − x 5 0 x 4 0 x ∗ 6 0 0 x 7 − x 6 0 0 x 4 − x ∗ 5 0 0 0 0 x 7 − x 6 x 5 x ∗ 4 3 7 7 7 7 7 7 7 7 5 (32) C ( x 0 , x 1 , x 2 , x 3 ) = 1 √ 2 ˆ − x ∗ 3 x ∗ 2 − x ∗ 1 − x 0 x ∗ 0 − x 1 − x 2 − x 3 ˜ T . (33) 8 Let i be any non-n egati ve integer . Deﬁne the f ollowing matrices: A (2 i ) = A ( x 8 i , x 8 i +1 , x 8 i +2 , x 8 i +3 ) , A (2 i + 1) = B ( x 8 i +4 , x 8 i +5 , x 8 i +6 , x 8 i +7 ) , (34) A ( i ) = C ( x 4 i , x 4 i +1 , x 4 i +2 , x 4 i +3 ) . One can easily verify that the matrices gi ven b y  A (0) A (1) A (1) A (0)  ,  A (1) − A (0) A (0) A (1)  , (35) are scaled CODs as the colu mns of the matrices are ortho gonal to each other and th e n orm of each colu mn is eq ual to square root of the sum of the nor ms of the variables of the design . By relabeling the variables in the matr ices A (0) , A (1) , A (0) and A (1) , it follows th at  A ( i ) A ( j ) A ( j ) A ( i )  (36) is a scaled COD whenever ( i + j ) is od d and  A ( i ) − A ( j ) A ( j ) A ( i )  , (37) is a scaled COD for all values o f i and j , i 6 = j . Let n be an integer such that n ≥ 9 . W e construct a ma trix ( D R ) n of size ν ( n ) × n as follows: L et t = n − 8 and W t and ˆ W t are the two rate- 1 R ODs of size [ ν ( t ) , t, ν ( t )] in ν ( t ) real variables y 0 , y 1 , · · · , y ν ( t ) − 1 constructed in th e previous subsection. Let H t and ˆ H t be the matrices formed by sub stituting y i with A (2 i + 1) in the ma trix W t and A (2 i ) in the matrix ˆ W t respectively for i = 0 to ν ( t ) − 1 . Note that the size of both H t and ˆ H t is 8 ν ( t ) × t . Let u = ν ( n ) / 8 . Let E 8 and O 8 , each of size 4 u × 8 , be deﬁned as follows: E 8 =         A (0) A (2) . . . A ( u − 2 )         O 8 =         A (1) A (3) . . . A ( u − 1)         . (38) Deﬁne the matrix ( D R ) n as ( D R ) n =  E 8 H t O 8 ˆ H t  . (39) Note th at the numb er o f rows and co lumns of the matr ix ( D R ) n are 16 · ν ( n − 8) = 8 · ν ( n ) / 8 = ν ( n ) and t + 8 = n respectively . T he following theo rem is the main result of this paper . Theor e m 5: Let n be a positiv e integer and ( D R ) n be the matrix as d eﬁned in (39). Then ( DR ) n is a rate- 1/2 scaled COD of size [ ν ( n ) , n, ν ( n ) 2 ] . Pr oof: For n ≤ 8 , one can c onstruct rate- 1/2 COD of size [ ν ( n ) , n, ν ( n ) 2 ] from a COD of size [8 , 8 , 4] gi ven in (3 1). W e assume that n ≥ 9 . Let p = ν ( n ) . W e have ( D R ) H n ( D R ) n =  E H 8 E 8 + O H 8 O 8 E H 8 H t + O H 8 ˆ H t H H t E 8 + ˆ H H t O 8 H H t H t + ˆ H H t ˆ H t  . From the construction of E 8 and O 8 giv en in (38), we hav e E H 8 E 8 + O H 8 O 8 = ( | x 0 | 2 + · · · + | x p 2 − 1 | 2 ) I 8 . From equation (37), we have H H t H t + ˆ H H t ˆ H t = ( | x 0 | 2 + · · · + | x p/ 2 − 1 | 2 ) I n − 8 , Thus it is enou gh to show that E H 8 H t + O H 8 ˆ H t = 0 8 × ( n − 8) where 0 8 × ( n − 8) is a matrix of size 8 × ( n − 8 ) co ntaining zero only . Let the j -th co lumn o f H t and ˆ H t be H t ( j ) a nd ˆ H t ( j ) respectively . Th en we show that Z ( j ) = E H 8 H t ( j ) + O H 8 ˆ H t ( j ) = 0 8 × 1 for all j ∈ { 0 , 1 , · · · , n − 8 − 1 } . Let u = p/ 8 . For con venien ce, we write γ for γ ν ( t ) . W e have E H 8 =  A H (0) A H (2) · · · A H ( u − 2)  , O H 8 =  A H (1) A H (3) · · · A H ( u − 1 )  , H t ( j ) =             s (0 , j ) A (2(0 ⊕ γ ( j )) + 1) s (1 , j ) A (2(1 ⊕ γ ( j )) + 1) . . s ( i, j ) A (2( i ⊕ γ ( j )) + 1) . . s ( u 2 − 1 , j ) A (2  ( u 2 − 1) ⊕ γ ( j )  + 1)             , ˆ H t ( j ) =             ˆ s (0 , j ) A (2(0 ⊕ γ ( j ))) ˆ s (1 , j ) A (2(1 ⊕ γ ( j ))) . . ˆ s ( i, j ) A (2( i ⊕ γ ( j ))) . . ˆ s ( u 2 − 1 , j ) A (2(( u 2 − 1) ⊕ γ ( j )))             , where s ( i, j ) and ˆ s ( i, j ) are deﬁned in (2 3) and ( 25) respec- ti vely . W e have Z ( j ) = u 2 − 1 X i =0 s ( i, j ) A H (2 i ) A (2( i ⊕ γ ( j )) + 1) + u 2 − 1 X i =0 ˆ s ( i, j ) A H (2 i + 1) A (2( i ⊕ γ ( j ))) . Now u 2 − 1 X i =0 ˆ s ( i, j ) A H (2 i + 1) A (2( i ⊕ γ ( j ))) = u 2 − 1 X i =0 ˆ s ( i ⊕ γ ( j ) , j ) A H (2( i ⊕ γ ( j )) + 1) A (2 i )  and s ( i, j ) = ˆ s ( i ⊕ γ ( j ) , j ) . 9 Hence Z ( j ) = u 2 − 1 X i =0  s ( i, j ) A H (2 i ) A (2( i ⊕ γ ( j )) + 1) + ˆ s ( i ⊕ γ ( j ) , j ) A H (2( i ⊕ γ ( j )) + 1) A (2 i )  = u 2 − 1 X i =0 s ( i, j )( A H (2 i ) A (2( i ⊕ γ ( j )) + 1) + A H (2( i ⊕ γ ( j )) + 1) A (2 i )) = 0 8 × 1 using (36). W e illustrate our main result in the fo llowing example. Example 2: For 9 tran smit an tennas, we have rate- 1 / 2 scaled COD of size [16 , 9 , 8 ] given by 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 x 0 − x ∗ 1 − x ∗ 2 0 − x ∗ 3 0 0 0 − x ∗ 7 √ 2 x 1 x ∗ 0 0 − x ∗ 2 0 − x ∗ 3 0 0 x ∗ 6 √ 2 x 2 0 x ∗ 0 x ∗ 1 0 0 − x ∗ 3 0 − x ∗ 5 √ 2 0 x 2 − x 1 x 0 0 0 0 − x ∗ 3 − x 4 √ 2 x 3 0 0 0 x ∗ 0 x ∗ 1 x ∗ 2 0 x ∗ 4 √ 2 0 x 3 0 0 − x 1 x 0 0 x ∗ 2 − x 5 √ 2 0 0 x 3 0 − x 2 0 x 0 − x ∗ 1 − x 6 √ 2 0 0 0 x 3 0 − x 2 x 1 x ∗ 0 − x 7 √ 2 x 4 − x ∗ 5 − x ∗ 6 − x ∗ 7 0 0 0 0 − x ∗ 3 √ 2 x 5 x ∗ 4 0 0 − x ∗ 6 − x ∗ 7 0 0 x ∗ 2 √ 2 x 6 0 x ∗ 4 0 x ∗ 5 0 − x ∗ 7 0 − x ∗ 1 √ 2 0 x 6 − x 5 0 x 4 0 0 − x ∗ 7 − x 0 √ 2 x 7 0 0 x ∗ 4 0 x ∗ 5 − x ∗ 7 0 x ∗ 0 √ 2 0 x 7 0 − x 5 0 x 4 0 x ∗ 6 − x 1 √ 2 0 0 x 7 − x 6 0 0 x 4 − x ∗ 5 − x 2 √ 2 0 0 0 0 x 7 − x 6 x 5 x ∗ 4 − x 3 √ 2 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 . (40) while the known rate 1 / 2 scaled COD for 9 transmit ante nna is giv en by [7] 1 √ 2 · 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 x 0 − x 1 − x 2 − x 3 − x 4 − x 5 − x 6 − x 7 − x 8 x 1 x 0 x 3 − x 2 x 5 − x 4 − x 7 x 6 x 9 x 2 − x 3 x 0 x 1 x 6 x 7 − x 4 − x 5 x 10 x 3 x 2 − x 1 x 0 x 7 − x 6 x 5 − x 4 x 11 x 4 − x 5 − x 6 − x 7 x 0 x 1 x 2 x 3 x 12 x 5 x 4 − x 7 x 6 − x 1 x 0 − x 3 x 2 x 13 x 6 x 7 x 4 − x 5 − x 2 x 3 x 0 − x 1 x 14 x 7 − x 6 x 5 x 4 − x 3 − x 2 x 1 x 0 x 15 x 8 − x 9 − x 10 − x 11 − x 12 − x 13 − x 14 − x 15 x 0 x 9 x 8 − x 11 x 10 − x 13 x 12 x 15 − x 14 − x 1 x 10 x 11 x 8 − x 9 − x 14 − x 15 x 12 x 13 − x 2 x 11 − x 10 x 9 x 8 − x 15 x 14 − x 13 x 12 − x 3 x 12 x 13 x 14 x 15 x 8 − x 9 − x 10 − x 11 − x 4 x 13 − x 12 x 15 − x 14 x 9 x 8 x 11 − x 10 − x 5 x 14 − x 15 − x 12 x 13 x 10 − x 11 x 8 x 9 − x 6 x 15 x 14 − x 13 − x 12 x 11 x 10 − x 9 x 8 − x 7 x ∗ 0 − x ∗ 1 − x ∗ 2 − x ∗ 3 − x ∗ 4 − x ∗ 5 − x ∗ 6 − x ∗ 7 − x ∗ 8 x ∗ 1 x ∗ 0 x ∗ 3 − x ∗ 2 x ∗ 5 − x ∗ 4 − x ∗ 7 x ∗ 6 x ∗ 9 x ∗ 2 − x ∗ 3 x ∗ 0 x ∗ 1 x ∗ 6 x ∗ 7 − x ∗ 4 − x ∗ 5 x ∗ 10 x ∗ 3 x ∗ 2 − x ∗ 1 x ∗ 0 x ∗ 7 − x ∗ 6 x ∗ 5 − x ∗ 4 x ∗ 11 x ∗ 4 − x ∗ 5 − x ∗ 6 − x ∗ 7 x ∗ 0 x ∗ 1 x ∗ 2 x ∗ 3 x ∗ 12 x ∗ 5 x ∗ 4 − x ∗ 7 x ∗ 6 − x ∗ 1 x ∗ 0 − x ∗ 3 x ∗ 2 x ∗ 13 x ∗ 6 x ∗ 7 x ∗ 4 − x ∗ 5 − x ∗ 2 x ∗ 3 x ∗ 0 − x ∗ 1 x ∗ 14 x ∗ 7 − x ∗ 6 x ∗ 5 x ∗ 4 − x ∗ 3 − x ∗ 2 x ∗ 1 x ∗ 0 x ∗ 15 x ∗ 8 − x ∗ 9 − x ∗ 10 − x ∗ 11 − x ∗ 12 − x ∗ 13 − x ∗ 14 − x ∗ 15 x ∗ 0 x ∗ 9 x ∗ 8 − x ∗ 11 x ∗ 10 − x ∗ 13 x ∗ 12 x ∗ 15 − x ∗ 14 − x ∗ 1 x ∗ 10 x ∗ 11 x ∗ 8 − x ∗ 9 − x ∗ 14 − x ∗ 15 x ∗ 12 x ∗ 13 − x ∗ 2 x ∗ 11 − x ∗ 10 x ∗ 9 x ∗ 8 − x ∗ 15 x ∗ 14 − x ∗ 13 x ∗ 12 − x ∗ 3 x ∗ 12 x ∗ 13 x ∗ 14 x ∗ 15 x ∗ 8 − x ∗ 9 − x ∗ 10 − x ∗ 11 − x ∗ 4 x ∗ 13 − x ∗ 12 x ∗ 15 − x ∗ 14 x ∗ 9 x ∗ 8 x ∗ 11 − x ∗ 10 − x ∗ 5 x ∗ 14 − x ∗ 15 − x ∗ 12 x ∗ 13 x ∗ 10 − x ∗ 11 x ∗ 8 x ∗ 9 − x ∗ 6 x ∗ 15 x ∗ 14 − x ∗ 13 − x ∗ 12 x ∗ 11 x ∗ 10 − x ∗ 9 x ∗ 8 − x ∗ 7 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 . where the decoding delay is 32 . For 10 transmit antenna s, the rate-1/2 scaled COD given by T arokh et al [7] of size [6 4 , 10 , 3 2 ] is g i ven in Appendix C, while the new rate- 1 / 2 code of size [32 , 10 , 1 6] is given by (41). 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 x 0 − x ∗ 1 − x ∗ 2 0 − x ∗ 3 0 0 0 − x ∗ 7 √ 2 − x ∗ 15 √ 2 x 1 x ∗ 0 0 − x ∗ 2 0 − x ∗ 3 0 0 x ∗ 6 √ 2 x ∗ 14 √ 2 x 2 0 x ∗ 0 x ∗ 1 0 0 − x ∗ 3 0 − x ∗ 5 √ 2 − x ∗ 13 √ 2 0 x 2 − x 1 x 0 0 0 0 − x ∗ 3 − x 4 √ 2 − x 12 √ 2 x 3 0 0 0 x ∗ 0 x ∗ 1 x ∗ 2 0 x ∗ 4 √ 2 x ∗ 12 √ 2 0 x 3 0 0 − x 1 x 0 0 x ∗ 2 − x 5 √ 2 − x 13 √ 2 0 0 x 3 0 − x 2 0 x 0 − x ∗ 1 − x 6 √ 2 − x 14 √ 2 0 0 0 x 3 0 − x 2 x 1 x ∗ 0 − x 7 √ 2 − x 15 √ 2 x 8 − x ∗ 9 − x ∗ 10 0 − x ∗ 11 0 0 0 − x ∗ 15 √ 2 x ∗ 7 √ 2 x 9 x ∗ 8 0 − x ∗ 10 0 − x ∗ 11 0 0 x ∗ 14 √ 2 − x ∗ 6 √ 2 x 10 0 x ∗ 8 x ∗ 9 0 0 − x ∗ 11 0 − x ∗ 13 √ 2 x ∗ 5 √ 2 0 x 10 − x 9 x 8 0 0 0 − x ∗ 11 − x 12 √ 2 x 4 √ 2 x 11 0 0 0 x ∗ 8 x ∗ 9 x ∗ 10 0 x ∗ 12 √ 2 − x ∗ 4 √ 2 0 x 11 0 0 − x 9 x 8 0 x ∗ 10 − x 13 √ 2 x 5 √ 2 0 0 x 11 0 − x 10 0 x 8 − x ∗ 9 − x 14 √ 2 x 6 √ 2 0 0 0 x 11 0 − x 10 x 9 x ∗ 8 − x 15 √ 2 x 7 √ 2 x 4 − x ∗ 5 − x ∗ 6 − x ∗ 7 0 0 0 0 − x ∗ 3 √ 2 x ∗ 11 √ 2 x 5 x ∗ 4 0 0 − x ∗ 6 − x ∗ 7 0 0 x ∗ 2 √ 2 − x ∗ 10 √ 2 x 6 0 x ∗ 4 0 x ∗ 5 0 − x ∗ 7 0 − x ∗ 1 √ 2 x ∗ 9 √ 2 0 x 6 − x 5 0 x 4 0 0 − x ∗ 7 − x 0 √ 2 x 8 √ 2 x 7 0 0 x ∗ 4 0 x ∗ 5 − x ∗ 7 0 x ∗ 0 √ 2 − x ∗ 8 √ 2 0 x 7 0 − x 5 0 x 4 0 x 6 − x 1 √ 2 x 10 √ 2 0 0 x 7 − x 6 0 0 x 4 − x ∗ 5 − x 2 √ 2 x 10 √ 2 0 0 0 0 x 7 − x 6 x 5 x ∗ 4 − x 3 √ 2 x 11 √ 2 x 12 − x ∗ 13 − x ∗ 14 − x ∗ 15 0 0 0 0 − x ∗ 11 √ 2 − x ∗ 3 √ 2 x 13 x ∗ 12 0 0 − x ∗ 14 − x ∗ 15 0 0 x ∗ 10 √ 2 x ∗ 2 √ 2 x 14 0 x ∗ 12 0 x ∗ 13 0 − x ∗ 15 0 − x ∗ 9 √ 2 − x ∗ 1 √ 2 0 x 14 − x 13 0 x 12 0 0 − x ∗ 15 − x 8 √ 2 − x 0 √ 2 x 15 0 0 x ∗ 12 0 x ∗ 13 − x ∗ 15 0 x ∗ 8 √ 2 x ∗ 0 √ 2 0 x 15 0 − x 13 0 x 12 0 x 14 − x 9 √ 2 − x 1 √ 2 0 0 x 15 − x 14 0 0 x 12 − x ∗ 13 − x 10 √ 2 − x 2 √ 2 0 0 0 0 x 15 − x 14 x 13 x ∗ 12 − x 11 √ 2 − x 3 √ 2 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 . (41) D. Ma ximal rate of th e scaled CODs It has been sh own by Liang [1] that th e maximal rate of a COD f or n tran smit an tennas is 1 2 + 1 2 t when n = 2 t − 1 or 2 t . The following result says when there is scalin g of atleast one column then the maximal rate is 1 2 . Theor e m 6: The maximal rate of a scaled COD, with scal- ing of at least one column, for n transmit antenn as is 1 2 . Pr oof: Let ( D R ) n be a scaled COD for n transmit antennas and there exists at-least one column of the m atrix such that when ev er a variable app ears in that co lumn, it is scaled by 1 √ 2 . Sinc e all the variables appe aring in a colu mn is scaled by 1 √ 2 , each variable must appea r twice in tha t co lumn. Let the n umber of distinct complex variables in ( D R ) n is k . Then 2 k ≤ p , i.e., k /p ≤ 1 / 2 . Thus if o ne allows to incorp orate a factor of 1 √ 2 for a ll the entr ies of any co lumn, ther e won’t be any improvement in the rate, on th e oth er han d, as we will have obser ved, we can construct some rate 1/2 codes with lesser decod ing delay . 10 I I I . D E L AY - M I N I M A L I T Y F O R 9 T R A N S M I T A N T E N NA S In this section, it is shown that the low-delay r ate 1 / 2 COD developed in the pr evious section is of m inimal delay for 9 transmit antenn as. T o prove this, we need some prelim inary facts regarding th e interrelation ship between r eal and comp lex ODs and certain bilinear maps. The ROD and bilin ear maps are intima tely r elated in the sense th at a R OD of size [ p, n, k ] exists if an d only if the re exists a norm ed bilinear map of the same size. The normed bilinear maps h av e been studied extensi vely and one ﬁnd a good intro duction to this top ic in the b ook by Shapiro [18]. As th e resu lts from the theory o f normed bilinear maps is used to prove ou r claim , these maps have b een deﬁned below and some facts are stated regard ing these maps. A bilinear map f (over a ﬁeld F ) is a map f : F k × F n → F p (42) ( x, y ) 7→ f ( x, y ) (43) such that it is linear in bo th x and y , i.e., f ( x 1 + x 2 , y ) = f ( x 1 , y ) + f ( x 2 , y ) an d f ( x, y 1 + y 2 ) = f ( x, y 1 ) + f ( x, y 2 ) for all x, x 1 , x 2 ∈ F k and y , y 1 , y 2 ∈ F n . T he space F p is called the target space of f . If the vector spaces und er considera tion are inne r product sp aces, for example, when the ﬁeld is re al number s o r com plex n umbers, the Euclid ean norm o f a vector x is d enoted by k x k . If a bilinear map preserves the n orm, then it is called a norm ed bilinear ma p. More precisely , Deﬁnition 3: A n ormed r eal bilinear ma p (NRBM) of size [ p, n, k ] is a map f : R k × R n → R p such that f is bilinear and n ormed i.e., k f ( x, y ) k = k x kk y k ∀ x ∈ R k , y ∈ R n . If f ( x, y ) = 0 imp lies x = 0 or y = 0 , th en such a map is called a nonsingu lar map . The following theorem gives a lower b ound on p for ﬁxed values of n an d k . Theor e m 7 (Hopf-Stiefel Theor em [18]): If there exists a nonsingu lar bilinear map o f size [ p, n, k ] over R , th en ( x + y ) p = 0 in the ring F 2 [ x, y ] / ( x n , y k ) . Deﬁnition 4: Let n, k be positive integers. Then the thr ee quantities n ◦ k , p B L and p N B L are deﬁned by • n ◦ k = min { p : ( x + y ) p = 0 in F 2 [ x, y ] / ( x n , y k ) } , • p B L ( n, k ) = min { p : ther e is a nonsing ular bilinear map [ p, n, k ] over R } , • p N B L ( n, k ) = min { p : there is a n ormed bilinea r map [ p, n, k ] over R } , The fo llowing basic facts about these quantities are well known [1 8]. p N B L ( n, k ) ≥ p B L ( n, k ) ≥ n ◦ k and p N B L ( n, k ) = n if and only if k ≤ ρ ( n ) where ρ is the Hurwitz- Radon function. It follows f rom the deﬁnition of n ◦ k that Pr oposition 1 ([18]): n ◦ k is a commutative binary op era- tion. ( I ) If k ≤ l then n ◦ k ≤ n ◦ l ( I I ) n ◦ k = 2 m if and only if k , n ≤ 2 m and k + n > 2 m . ( I I I ) If n ≤ 2 m then n ◦ ( k + 2 m ) = n ◦ k + 2 m . Example 3: L et us compute 10 ◦ 1 0 . W e observe that 1 0 < 2 4 , but (1 0 + 10) > 16 . So, 10 ◦ 1 0 = 16 . The r elation between RODs and NRBMs has b een o bserved by W ang an d Xia in [17]. For th e sake of co mpleteness and since th e p roof of this fact gives the e x plicit relation between the NRBM and the row-vector rep resentation matrices of the R OD wh ich is in correspo ndence with it, we give here the proof of the following lemm a. Lemma 4: A R OD of size [ p, n, k ] exists if an d only if th ere exists a normed real bilinear map of size [ p, n, k ] . Pr oof: Let x = ( x 1 , x 2 , ..., x k ) T , y = ( y 1 , · · · , y n ) T and z = ( z 1 , · · · , , z p ) T be real column vectors. Let A be a R OD of size [ p, n, k ] gi ven by A = P k i =1 A i x i where the p × n real matrices are the d ispersion matrices or weight matrices deﬁning the design A [1]. Let f : R k × R n → R p ( x , y ) 7→ ( k X i =1 A i x i ) y . W e show that f is a nor med r eal bilinear map o f size [ p , n, k ] . Let z = f ( x, y ) . Th en z i = x T B i y wh ere the k × n real matrices B i , i = 1 , 2 , · · · , p, are the row vector representatio n [1] of the d esign A. This representation shows that f is bilinear . Moreover , f is normed, since k f ( x , y ) k 2 = k Ay k 2 = ( Ay ) T Ay = y T ( x 2 1 + x 2 2 + ... + x 2 k ) I n ) y = k x k 2 k y k 2 . T o sho w that the converse ho lds, let f be the normed bilinear map given by f : R k × R n → R p ( x , y ) 7→ z . As f is linear in both x and y , we have z = Ay wher e A is an p × n matrix where each entry of th e matrix is a r eal lin ear combinatio n of the v ar iables x 1 , · · · , x k . As f is normed, we have k z k 2 = k f ( x, y ) k 2 = k x k 2 k y k 2 . But f ( x, y ) = Ay . Then, k Ay k 2 = (( x 2 1 + · · · + x 2 k ) I n ) y T y . So, we have y T Ay = (( x 2 1 + · · · + x 2 k ) I n ) y T y . As y consists of variables, the above equ ation is equiv alent to A T A = ( x 2 1 + x 2 2 + ... + x 2 k ) I n . W e now p rove the main resu lt of this section. Theor e m 8: The m inimum decoding delay o f the rate- 1 / 2 COD admitting linear combin ation of com plex variables for 9 transmit antennas is 16 . Pr oof: In this proo f, we assume that the R OD and COD admit linear com bination o f two or mor e variables as its entries. Su ppose, there exists a COD of size [2 x, 9 , x ] where x is an in teger less than 8 . Th is imp lies existence of a ROD of size [4 x, 18 , 2 x ] , x < 8 which is obta ined b y r eplacing each complex entry by its 2 × 2 real matrix represen tation. In the remaining part of the proo f, we show that such a R O D does not exist thus proving the th eorem. If a ROD of size [4 x, 18 , 2 x ] exists, then there also exists a normed biline ar map of the same size by Le mma 4 which implies that 4 x ≥ 18 ◦ 2 x . As 18 ◦ 2 x ≥ 18 , w e have 4 x ≥ 18 i.e., x ≥ 5 for x being an in teger . Thus we have thr ee p ossible choices for x , namely 5 , 6 and 7 . By Proposition 1, we ha ve 18 ◦ 2 x = 2 6 , 28 , 30 for x = 5 , 6 and 7 respectively . I n all the cases, 18 ◦ 2 x > 4 x which con tradicts the fact that 4 x ≥ 18 ◦ 2 x . 11 I V . P A P R R E D U C T I O N O F R A T E - 1 / 2 S C A L E D C O D S In this sectio n, we study the P APR p roperties of Scaled CODs constructed in th is paper . Note that in the construction of T J C n giv en in [7], ev en th ough the d elay is mo re, there are no zero entries in the design matrix . On the contr ary , in ou r construction of low-day codes ( D R ) n there are zero entr ies. T o b e spe ciﬁc, observe that the ﬁr st eigh t column s of rate- 1 / 2 code ( DR ) n , n ≥ 9 giv en in (3 9) con tains as many zero as the nu mber of non- zero entr ies in it, w hile there is no zero in the remainin g c olumns of the m atrix. When the n umber of transmit antennas n is more than 7 , th e total number of zeros in th e codew o rd matr ix is eq ual to 8( ν ( n ) / 2) = 4 ν ( n ) . Hence the fraction of zer os in the codeword matrix is equ al to 4 ν ( n ) nν ( n ) = 4 / n for n ≥ 8 . Now in the rem aining p art of th is section, we show that o ne can further reduc e the number of zeros in ( D R ) n by suitab ly choosing a post-multiplication matrix to it with out increasing signaling complexity [15] of the code. As seen easily , on ly the ﬁrst eigh t column co ntain zeros while the o thers do not. Moreover, the z eros in 0 -th colum n and the 7 − th column occupy comp lementary locations, so is also f or th e p airs o f colum ns given b y (1 , 6) , (2 , 5) a nd (3 , 4 ) . What it essentially sugge sts is that we can pe rform som e elementary column ope rations wh ich will result in a code in which all the e ntries a re non -zero. In other words, if the r ate- 1/2 COD is of size p × n , th en we p ost-multiply it with a matrix Q n of size n × n given by Q n =  A 0 0 I n − 8  where A is a matrix of size 8 × 8 given by A = 1 √ 2             1 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 1 − 0 0 0 0 0 1 0 0 − 0 0 0 1 0 0 0 0 − 0 1 0 0 0 0 0 0 −             and I n − 8 is the ( n − 8) × ( n − 8) identity matrix, the n all the entries of the scaled COD gi ven by ( DR ) n Q n , are non-zero. W e for mally present this fact as: Theor e m 9: The matrix Q n when p ost-multiplied with a rate-1/2 scaled COD ( D R ) n giv en by ( D R ) n =  E 8 H t O 8 ˆ H t  . (44) always g iv es a COD with no zeros. Mo reover , th e matr ix Q n does not dep end on any particu lar constru ction proc edure (namely the m aps γ t and ψ t ) used to obtain the con stituent rate-1 R ODs. Pr oof: It is clear that the ﬁrst 8 columns o f the matrix has 50% zeros in it and in th e remaining n − 8 colum ns formed by H t and ˆ H t , there ar e n o zer os as bo th these matr ices ar e constructed from rate-1 R OD by substituting all the variables in it with app ropriate 8 -tup le colum n vectors. Here ne ither rate-1 ROD nor th e 8 - tuple column vector has any any zero in it. Theref ore, the matrix Q n giv es a ra te 1 / 2 scales COD without any zeros irr espectiv e of h ow th e rate-1 R ODs are obtained for the construction of ( D R ) n . Example 4: T he ra te-1/2 cod e with no zero entry fo r 9 transmit antennas is given by 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 y 0 − y ∗ 1 − y ∗ 2 − y ∗ 3 y ∗ 3 − y ∗ 2 − y ∗ 1 y 0 − y ∗ 7 y 1 y ∗ 0 − y ∗ 3 − y ∗ 2 − y ∗ 2 y ∗ 3 y ∗ 0 y 1 y ∗ 6 y 2 − y ∗ 3 y ∗ 0 y ∗ 1 y ∗ 1 y ∗ 0 y ∗ 3 y 2 − y ∗ 5 − y ∗ 3 y 2 − y 1 y 0 y 0 − y 1 y 2 y ∗ 3 − y 4 y 3 y ∗ 2 y ∗ 1 y ∗ 0 − y ∗ 0 − y ∗ 1 − y ∗ 2 y 3 y ∗ 4 y ∗ 2 y 3 y 0 − y 1 y 1 − y 0 y 3 − y ∗ 2 − y 5 − y ∗ 1 y 0 y 3 − y 2 y 2 y 3 − y 0 y ∗ 1 − y 6 y ∗ 0 y 1 − y 2 y 3 y 3 y 2 − y 1 − y ∗ 0 − y 7 y 4 − y ∗ 5 − y ∗ 6 − y ∗ 7 − y ∗ 7 − y ∗ 6 − y ∗ 5 y 4 − y ∗ 3 y 5 y ∗ 4 − y ∗ 7 − y ∗ 6 y ∗ 6 y ∗ 7 y ∗ 4 y 5 y ∗ 2 y 6 − y ∗ 7 y ∗ 4 y ∗ 5 − y ∗ 5 y ∗ 4 y ∗ 7 y 6 − y ∗ 1 − y ∗ 7 y 6 − y 5 y 4 − y 4 − y 5 y 6 y ∗ 7 − y 0 y 7 y ∗ 6 y ∗ 5 y ∗ 4 y ∗ 4 − y ∗ 5 − y ∗ 6 y 7 y ∗ 0 y ∗ 6 y 7 y 4 − y 5 − y 5 − y 4 y 7 − y ∗ 6 − y 1 − y ∗ 5 y 4 y 7 − y 6 − y 6 y 7 − y 4 y ∗ 5 − y 2 y ∗ 4 y 5 − y 6 y 7 − y 7 y 6 − y 5 − y ∗ 4 − y 3 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 V . D I S C U S S I O N This pap er gives rate- 1 / 2 CODs fo r n tr ansmit anten nas with decodin g d elay eq ual to ν ( n ) . The d ecoding d elay of these codes is half of that of the rate-1/2 C ODs gi ven in T ar okh et al [7]. As the m aximal rate o f a scaled COD is close to 1 / 2 for large num ber of transm it antenn as, the cod es constru cted in this paper are b etter th an the co des constructed b y Liang [ 1] or Lu et al [10] wh en con sidered for large n umber of transmit antennas. Ano ther advantage with the de signs repo rted in this paper is that they do not co ntain z ero entrie s leading to low P APR. All the four co nstructions nam ely Adams, Lax and Phillips construction from Quatern ions, Octo nion, Geramita-Pullm an construction and the con struction given in this paper will give the same square R OD if nu mber of tran smit anten nas is less than or equal to 8 . Therefo re, these four construction will generate the same r ate 1 / 2 scaled COD if the numb er o f transmit antenna s ( of the scaled COD) is less than o r equal to 16 . For mo re than 16 an tennas, rate-1 /2 scaled CODs will vary with th e methods chosen for th e construction of rate-1 R ODs. Due to th e largen ess of the matrice s inv olved , it is not possible to display two distinct rate-1/2 scaled CODs f or 17 transmit a ntennas, obtained by two different constru ction proced ures for rate- 1 R ODs. It is not kn own whether the low-delay for rate 1 / 2 scaled CODs we have achieved is minimal delay excep t f or the case o f 9 tr ansmit anten nas. W e conjecture that ν ( n ) is th e minimum value of the decod ing delay of rate -1/2 scaled CODs for a ny n transmit antennas. It will be in teresting to see whether this is indeed true. 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Sundar Rajan, “Square Complex Orthogonal Designs with Low P A PR and Signaling Complexi ty , ” T o appear in IEEE Tra nsactio ns on W ireless Communications. [15] Sm araji t Das and B. Sundar Rajan, “Low-d elay , High-rate, Non-square STBCs from Scaled Complex Orthogonal Designs, ” Proceedin gs of IEEE Internat ional Symposium on Informat ion Theory , (ISIT 2008), T oronto, Canada, July 6-11, 2008, pp. 1483-1487. [16] C. Y uen, Y . L. Guan and T . T . Tjhung, “Power -Balanced Orthogonal Space-T ime Block Code, ” accepted for publication in IEEE Trans. V ehicul ar T echnolo gy . [17] Haiquan W ang and Xiang-Gen Xia, “Upper Bounds of Rates of Comple x Ortho gonal Space-Ti m e Block Codes, ” IEEE Trans. Inform. Theory , 49 (2003), pp. 2788-2796. [18] D. B. Shapiro, Compositio ns of Quadrati c forms, Berlin, Germany: W alte r de Gruyter , 2000. A P P E N D I X A R E C U R S I V E C O N S T RU C T I O N O F R t In this appendix we show that the R ODs R t can be constructed recursively . Let K t = B t for t = 1 , 2 , 4 and 8 . The four squ are ODs K t , t = 1 , 2 , 4 , 8 are shown below . ( x 0 ) , „ x 0 x 1 − x 1 x 0 « , 0 B @ x 0 x 1 x 2 x 3 − x 1 x 0 − x 3 x 2 − x 2 x 3 x 0 − x 1 − x 3 − x 2 x 1 x 0 1 C A , 0 B B B B B B B B B @ x 0 x 1 x 2 x 3 x 4 x 5 x 6 x 7 − x 1 x 0 − x 3 x 2 − x 5 x 4 x 7 − x 6 − x 2 x 3 x 0 − x 1 − x 6 − x 7 x 4 x 5 − x 3 − x 2 x 1 x 0 − x 7 x 6 − x 5 x 4 − x 4 x 5 x 6 x 7 x 0 − x 1 − x 2 − x 3 − x 5 − x 4 x 7 − x 6 x 1 x 0 x 3 − x 2 − x 6 − x 7 − x 4 x 5 x 2 − x 3 x 0 x 1 − x 7 x 6 − x 5 − x 4 x 3 x 2 − x 1 x 0 1 C C C C C C C C C A . (45) It follows th at K T t = K T t ( x 0 , x 1 , · · · , x t − 1 ) = K t ( x 0 , − x 1 , · · · , − x t − 1 ) and − K T t = K t ( − x 0 , x 1 , · · · , x t − 1 ) for t = 1 , 2 , 4 or 8 . The expression for R t of or der t as gi ven in Theor em 4 gives rise to the fo llowing recursi ve construction of R t . Giv en two matrices U = ( u ij ) of size v 1 × w 1 and V of size v 2 × w 2 , we deﬁne the Kr oneck er pr oduct or tensor pr oduc t of U and V as the following v 1 v 2 × w 1 w 2 matrix:      u 11 V u 12 V · · · u 1 w 1 V u 11 V u 12 V · · · u 1 w 1 V . . . . . . . . . . . . u v 1 1 V u v 1 2 V · · · u v 1 w 1 V      . Let I n be an identity matrix of size n . Deﬁne I 0 2 =  1 0 0 1  , I 1 2 =  1 0 0 − 1  , I 2 2 =  0 1 1 0  , I 3 2 =  0 − 1 1 0  , I 0 4 = I 4 , I 1 4 = I 3 2 ⊗ I 2 2 , I 0 8 = I 8 , I 1 8 = I 0 2 ⊗ I 1 4 , I 2 8 = I 3 2 ⊗ I 1 2 ⊗ I 2 2 , I 3 8 = I 3 2 ⊗ I 2 2 ⊗ I 0 2 . Let y 0 , · · · , y 5 be real variables. Deﬁne T 4 ( y 0 , y 1 ) = y 0 I 0 4 + y 1 I 1 4 , T 8 ( y 2 , y 3 , y 4 , y 5 ) = y 2 I 0 8 + y 3 I 1 8 + y 4 I 2 8 + y 5 I 3 8 . W e have f our R ODs o f order n = 2 a with a = 0 , 1 , 2 , 3 as giv en in (45) which are respecti vely K 1 , K 2 , K 4 and K 8 . Assuming that a square R OD of or der n = 2 4 l − 1 , l ≥ 1 R n = R n ( x 0 , · · · , x ρ ( n ) − 1 ) which has ρ ( n ) real variables, is given, then we construc t R 2 n , R 4 n , R 8 n , R 16 n of ord er 2 n , 4 n , 8 n an d 16 n respectively giv en by (4 6), as shown at the top of the next p age where y i = x ρ ( n )+2+ i and R T t = R T t ( x 0 , x 1 , · · · , x ρ ( t ) − 1 ) = R t ( x 0 , − x 1 , · · · , − x ρ ( t ) − 1 ) , − R T t = R t ( − x 0 , x 1 , · · · , x ρ ( t ) − 1 ) . A P P E N D I X B A D A M S - L A X - P H I L L I P S A N D G E R A M I T A - P U L L M A N C O N S T RU C T I O N S A S S P E C I A L C A S E S In th is appen dix we sh ow that the well kn own constructions of square R ODs b y Adams-Lax- Phillips using Octo nions and Quaternion s as well a s the constru ction b y Ge ramita and Pullman are nothing but our constructio n correspo nding to speciﬁc cho ices of the functio ns γ t and ψ t deﬁned by ( 7) and (8). I t turn s out to be conv e nient to use the m ap χ t = ψ t γ t instead of the map ψ t . Note that both γ t and χ t act on the set Z ρ ( t ) and are injec ti ve. Now given γ t and χ t , we have ψ t = χ t γ ( − 1) t . With this new d eﬁnition, we can reformu late the criteria gi ven in Theorem 4 as follows. | ( χ t ( x ) ⊕ χ t ( y )) · ( γ t ( x ) ⊕ γ t ( y )) | (47) is an odd integer ∀ x, y ∈ Z ρ ( t ) , x 6 = y . 13 R 2 n =  R n x ρ ( n ) I n − x ρ ( n ) I n R T n  , R 4 n =  R 2 n x ρ ( n )+1 I 2 n − x ρ ( n )+1 I 2 n R T 2 n  , R 8 n =  R 4 n T 4 ( y 0 , y 1 ) ⊗ I n T 4 ( − y 0 , y 1 ) ⊗ I n R T 4 n  , R 16 n =  R 8 n T 8 ( y 2 , y 3 , y 4 , y 5 ) ⊗ I n T 8 ( − y 2 , y 3 , y 4 , y 5 ) ⊗ I n R T 8 n  (46) In the f ollowing lemma, we deﬁne γ t and χ t in thre e d ifferent ways and these map s are shown to satisfy the relatio n giv en in (47). Alth ough b oth γ t and χ t are different for all th e three cases fo r arbitr ary values of t , γ t is the identity map wh en t = 1 , 2 , 4 or 8 . He nce χ t = ψ t if t ∈ { 1 , 2 , 4 , 8 } and is gi ven by (14). Lemma 5: L et t = 2 a , a = 4 c + d , m ∈ { 0 , 1 , · · · , 7 } . Let γ t and χ t be two m aps d eﬁned over Z ρ ( t ) in th ree different ways as g i ven below . Id entify γ t ( Z ρ ( t ) ) and χ t ( Z ρ ( t ) ) as subsets of F a 2 . Then | ( γ t ( x 1 ) ⊕ γ t ( x 2 )) · ( χ t ( x 1 ) ⊕ χ t ( x 2 )) | is odd for all x 1 , x 2 ∈ Z ρ ( t ) , x 1 6 = x 2 . For x = 8 l + m ∈ Z ρ ( t ) , (i) γ t (8 l + m ) = t (1 − 2 − l ) + 8 l m, χ t (8 l + m ) =          0 if l = 0 , m = 0 t. 2 − l if l 6 = 0 , m = 0 8 l χ 2 d ( m ) if l = c, m 6 = 0 t. 2 − l − 1 + 8 l χ 8 ( m ) if l 6 = c, m 6 = 0 , (ii) γ t (8 l + m ) = ( t (1 − 2 − 2 l ) + 2 2 l m if 0 ≤ m ≤ 3 t (1 − 2 − 2 l − 1 ) + 2 2 l ( m − 4) if 4 ≤ m ≤ 7 , χ t (8 l + m ) = 8 > > > > > > > > > < > > > > > > > > > : 0 if l = 0 , m = 0 t. 2 − 2 l if l 6 = 0 , m = 0 t. 2 − 2 l − 1 if l 6 = 0 , m = 4 4 if l = 0 , m = 4 2 2 l χ 2 d ( m ) if l = c, m 6 = 0 t. 2 − 2 l − 1 + 2 2 l χ 4 ( m ) if l 6 = c, m ∈ { 1 , 2 , 3 } t. 2 − 2 l − 2 + 2 2 l χ ′ 4 ( m − 4) if l 6 = c, m ∈ { 5 , 6 , 7 } , where χ ′ 4 =  0 1 2 3 0 1 3 2  . (iii) γ t (8 l + m ) = ( 8 t 15 (1 − 2 − 4 l ) + tm 16 l +1 if l < c , 8 t 15 (1 − 2 − 4 l ) + m if l = c χ t (8 l + m ) =          0 if l = 0 , m = 0 t 2 2 − 4( l − 1) if l 6 = 0 , m = 0 χ 2 d ( m ) if l = c, m 6 = 0 . t 2 2 − 4 l + tχ 8 ( m ) 2 4( l +1) if l 6 = c, m 6 = 0 . Pr oof: W e gi ve pro of o nly for the case (i). Th e cases (ii) and (iii) can be proved similarly . It is enough to prove th at (B1) | γ t ( x ) · χ t ( x ) | is odd for all x 6 = 0 , x ∈ Z ρ ( t ) and (B2) | γ t ( x 1 ) · χ t ( x 2 ) | + | γ t ( x 2 ) · χ t ( x 1 ) | is odd fo r all x 1 , x 2 ∈ Z ρ ( t ) , x 1 6 = x 2 , x 1 6 = 0 , x 2 6 = 0 . Let γ t (8 l + m ) = γ (1) t (8 l + m ) + γ (2) t (8 l + m ) such that γ (1) t (8 l + m ) = t (1 − 2 − l ) and γ (2) t (8 l + m ) = 8 l m . Similarly , let χ t (8 l + m ) = χ (1) t (8 l + m ) + χ (2) t (8 l + m ) such that χ (1) t (8 l + m ) =          0 if l = 0 , m = 0 , t 2 − l if l 6 = 0 , m = 0 , 0 if l = c, m 6 = 0 , t 2 − l − 1 if l 6 = c, m 6 = 0 , χ (2) t (8 l + m ) =          0 if l = 0 , m = 0 , 0 if l 6 = 0 , m = 0 , 8 l χ 2 d ( m ) if l = c, m 6 = 0 , 8 l χ 8 ( m ) if l 6 = c, m 6 = 0 . Let 8 l + m 6 = 0 and 8 l ′ + m ′ 6 = 0 . From the deﬁnition o f γ i t , χ i t , i = 1 , 2 , it follows that ( A 1) | χ (2) t (8 l + m ) · γ (2) t (8 l ′ + m ′ ) | = 0 if l 6 = l ′ , ( A 2) | χ (1) t (8 l + m ) · γ (1) t (8 l ′ + m ′ ) | = 1 if l < l ′ , ( A 3) | χ (1) t (8 l + m ) · γ (1) t (8 l ′ + m ′ ) | = 0 if l > l ′ or if l = l ′ , m 6 = 0 , ( A 4) | χ (1) t (8 l ) · γ (1) t (8 l + m ) | = 1 if l 6 = 0 , ( A 5) | χ (1) t ( x ) · γ (2) t ( y ) | = | χ (2) t ( x ) · γ (1) t ( y ) | = 0 ∀ x, y ∈ Z ρ ( t ) , ( A 6) | χ (2) t (8 l ) · γ (2) t (8 l + m ) | = | χ (2) t (8 l + m ) · γ (2) t (8 l ) | = 0 . First we prove (B1). Let x = 8 l + m with m 6 = 0 . W e hav e | χ t ( x ) · γ t ( x ) | ≡ | χ (1) t (8 l + m ) · γ (1) t (8 l + m ) | + | χ (2) t (8 l + m ) · γ (2) t (8 l + m ) | + | χ (1) t (8 l + m ) · γ (2) t (8 l + m ) | + | χ (2) t (8 l + m ) · γ (1) t (8 l + m ) | = | χ (1) t (8 l + m ) · γ (1) t (8 l + m ) | + | χ (2) t (8 l + m ) · γ (2) t (8 l + m ) | by (A5) = | χ (2) t (8 l + m ) · γ (2) t (8 l + m ) | using (A3) = | χ e ( m ) · m | , e = 2 d if l = c , else e = 8 But | χ e ( m ) · m | is an od d n umber by Lemma 2. If m = 0 , we have | γ t ( x ) · χ t ( x ) | = 1 by (A4) . T o prove (B2), let x 1 6 = 0 and x 2 6 = 0 . Write x 2 = 8 l 2 + m 2 , x 1 = 8 l 1 + m 1 with x 2 > x 1 . W e have two cases: (C1): l 2 > l 1 , (C2): l 2 = l 1 = l , m 2 > m 1 . Case (C1): we hav e χ t ( x 2 ) · γ t ( x 1 ) = χ (1) t (8 l 2 + m 2 ) · γ (1) t (8 l 1 + m 1 ) ⊕ χ (2) t (8 l 2 + m 2 ) · γ (2) t (8 l 1 + m 1 ) by (A5) . But | χ (1) t (8 l 2 + m 2 ) · γ (1) t (8 l 1 + m 1 ) | = 0 by (A3) and | χ (2) t (8 l 2 + m 2 ) · γ (2) t (8 l 1 + m 1 ) | = 0 by (A1), thus | χ t ( x 2 ) · γ t ( x 1 ) | = 0 . 14 Now χ t ( x 1 ) · γ t ( x 2 ) = χ (1) t (8 l 1 + m 1 ) · γ (1) t (8 l 2 + m 2 ) ⊕ χ (2) t (8 l 1 + m 1 ) · γ (2) t (8 l 2 + m 2 ) by (A5). But | χ (2) t (8 l 1 + m 1 ) · γ (2) t (8 l 2 + m 2 ) | = 0 by (A1 ) and | χ (1) t (8 l 1 + m 1 ) · γ (1) t (8 l 2 + m 2 ) | = 1 by (A2). Hence | χ t ( x 1 ) · γ t ( x 2 ) | + | χ t ( x 2 ) · γ t ( x 1 ) | is an o dd number . Case (C2): we consider two following cases: (i) m 1 6 = 0 an d (ii) m 1 = 0 . No te that m 2 is always n on-zero . Let d = | ( χ t ( x 1 ) · γ t ( x 2 )) ⊕ ( χ t ( x 2 ) · γ t ( x 1 )) | . Case (i): W e hav e d ≡ | χ (2) t (8 l + m 1 ) · γ (2) t (8 l + m 2 ) | + | χ (2) t (8 l + m 2 ) · γ (2) t (8 l + m 1 ) | by (A3) and (A5) = | ( χ e ( m 1 ) · m 2 ) ⊕ ( χ e ( m 2 ) · m 1 ) | , e = 2 d if l = c , else e = 8 which is an odd number by Lemma 2. Case (ii): Since m 1 = 0 , therefore l 6 = 0 . W e have d ≡ | χ (1) t (8 l ) · γ (2) t (8 l + m 2 ) | + | χ (1) t (8 l + m 2 ) · γ (1) t (8 l ) | by (A6). = 1 by (A3) and (A4 ). By Lemma 5 and Theo rem 3, the matrix B t deﬁned by the two fun ctions γ t and χ t is a square R OD in all the thr ee cases. W e refer to th ese three different R ODs by A t , ˆ A t and P t correspo nding to the pair of fu nctions deﬁned in (i), (ii) and (iii) respectively . Observe that o ur con struction R t is different from any of A t , ˆ A t and P t for general values of t. Now , we p roceed to show that the d esigns A t , ˆ A t and P t are essentially the Adams-L ax-Phillips co nstruction using Octo- nions and Quatern ions and th e Ge ramita-Pullman constructio n respectively with chan ge in sign of some rows o r columns. A. Adams-Lax-P hillips Construction fr om Octonion s as a sp e- cial case The Adams-L ax-Phillips co nstruction from Octonions is giv en by indu ction from order n = 2 a to 16 n as follows [1]: Denoting the square ROD of order n = 2 a resulting from the Adams-Lax- Phillips constru ction using Octonions by O n = O n ( x 0 , · · · , x ρ ( n ) − 1 ) which has ρ ( n ) real variables, the squar e R OD of ord er 16 n with ( ρ ( n ) + 8 ) real v a riables x i , i = 0 , 1 , · · · , ρ ( n ) + 7 , O 16 n = O 16 n ( x 0 , · · · , x ρ ( n )+7 ) is giv en by O 16 n =  I n ⊗ K 8 ( y 0 , · · · , y 7 ) O n ⊗ I 8 O T n ⊗ I 8 I n ⊗ ( − K T 8 ( y 0 , · · · , y 7 ))  with y i = x ρ ( n )+ i . W ith re-arran gement of variables and change in signs, we rewrite the design O 16 n as O ( O ) 16 n = " I n ⊗ K 8 ( x 0 , · · · , x 7 ) O ( O ) n ( y 0 , · · · , y ρ ( n ) − 1 ) ⊗ I 8 − O ( O ) T n ( y 0 , · · · , y ρ ( n ) − 1 ) ⊗ I 8 I n ⊗ K T 8 ( x 0 , · · · , x 7 ) # (49) with y i = x 8+ i and O ( O ) n = O n , n = 1 , 2 , 4 , 8 . The reason why we con sider this rearr anged version is that we show in Lemma 6 that A t is same as O ( O ) 2 n with t = 16 n. Lemma 6: L et t ≥ 16 and a power of 2 . Also, let A t be the squa re R OD of order t as given in L emma 5 (i), an d O ( O ) 16 n be the squ are R OD given in (4 9) which is of o rder 16 n . Then A t = O ( O ) 16 n for t = 16 n . Pr oof: W e prove it by ind uction on t . For t = 1 , 2 , 4 an d 8 , A t = K t and the COD O ( O ) t of order t is also g i ven b y K t . Hence the lemma hold s fo r t = 1 , 2 , 4 and 8 . Assuming that the lemm a ho lds for t = n , i.e., A n = O ( O ) n of o rder n , we ha ve to prove that the lemma also holds for t = 16 n, i.e ., A 16 n = O ( O ) 16 n . Let A 16 n =  ˆ A 11 ˆ A 12 ˆ A 21 ˆ A 22  (50) where ˆ A ij , 1 ≤ i, j ≤ 2 are square matrice s of size 8 n × 8 n . It is ea sy to check that the location of non-ze ro variables in the matrix A 16 n coincide with that of O ( O ) 16 n . Ther efore it is enoug h to show the signs ( positiv e/negativ e polar ity) of the correspo nding entr ies in the tw o designs are same i.e., 1) µ 16 n ( i, j ) = µ 16 n ( i %8 , j %8) for 0 ≤ i , j ≤ 8 n − 1 , 2) µ 16 n ( i, j ) = µ 8 ( i, j ) for 0 ≤ i, j ≤ 7 , 3) µ 16 n ( i, j ) = µ 16 n ( i ⊕ i %8 , j ⊕ j %8) if 0 ≤ i ≤ 8 n − 1 , 8 n ≤ j ≤ 16 n − 1 , 4) µ 16 n (8 i, 8 n ⊕ 8 j ) = µ n ( i, j ) for 0 ≤ i, j ≤ n − 1 , 5) µ 16 n (8 n ⊕ i, 8 n ⊕ j ) = µ 16 n ( i, j ) if i ⊕ j = 0 or i ⊕ j > 8 n , 6) µ 16 n (8 n ⊕ i, 8 n ⊕ j ) = − µ 16 n ( i, j ) if i ⊕ j ∈ { 1 , 2 , · · · , 7 } ∪ { 8 n } . Note that 1) & 2) together imply ˆ A 11 = I n ⊗ K 8 ( x 0 , · · · , x 7 ) , 3) & 4) together imply ˆ A 12 = O ( O ) n ⊗ I 8 and 5) & 6) together imply ˆ A 22 = A T 11 , ˆ A 21 = − A T 12 . Let A 16 n ( i, j ) 6 = 0 . Then i ⊕ j ∈ ˆ Z ρ (16 n ) and µ 16 n ( i, j ) = ( − 1) | i · ψ 16 n ( i ⊕ j ) | . T o prove 1), we have to sho w that | i · ψ 16 n ( i ⊕ j ) | ≡ | ( i %8) · ψ 16 n ( i %8 ⊕ j %8) for 0 ≤ i, j ≤ 8 n − 1 . W e have i ⊕ j = (16 n )(1 − 2 − l ) + 8 l m and i ⊕ j < 8 n . So l = 0 and i ⊕ j = m . i.e., i ⊕ j = i %8 ⊕ j %8 . Thus it is en ough to prove that | ( i ⊕ i %8) · ψ 16 n ( i ⊕ j ) | ≡ 0 Now ( i ⊕ i %8) < 8 n , 8 divides ( i ⊕ i %8) and ψ 16 n ( i ⊕ j ) = 8 n ⊕ ψ 8 ( m ) , hence the statement hold s. The statemen t 2) is true as | i · ψ 16 n ( i ⊕ j ) | ≡ | i · ψ 8 ( i ⊕ j ) | for 0 ≤ i, j ≤ 7 . In order to prove 3), we mu st hav e | i · ψ 16 n ( i ⊕ j ) | ≡ | ( i ⊕ i %8) · ψ 16 n (( i ⊕ i %8) ⊕ ( j ⊕ j %8)) | i.e., | ( i %8 ) · ψ 16 n (( i ⊕ i %8) ⊕ ( j ⊕ j %8)) | ≡ 0 . As 8 n ≤ i ⊕ j ≤ 16 n − 1 , we have i ⊕ j = (16 n )(1 − 2 − l ) + 8 l m with 15 O 3 = 0 B B @ I n ⊗ L 4 ( x 0 , x 1 , x 2 , x 3 ) 0 4 n I n ⊗ R 4 ( x 4 , x 5 , x 6 , x 7 ) O 1 ( y 0 , · · · , y ρ ( n ) − 1 ) ⊗ I 4 0 4 n I n ⊗ L 4 ( x 0 , x 1 , x 2 , x 3 ) − O T 1 ( y 0 , · · · , y ρ ( n ) − 1 ) ⊗ I 4 I n ⊗ R T 4 ( x 4 , x 5 , x 6 , x 7 ) I n ⊗ − R T 4 ( x 4 , x 5 , x 6 , x 7 ) O 1 ( y 0 , · · · , y ρ ( n ) − 1 ) ⊗ I 4 I n ⊗ L T 4 ( x 0 , x 1 , x 2 , x 3 ) 0 4 n − O T 1 ( y 0 , · · · , y ρ ( n ) − 1 ) ⊗ I 4 I n ⊗ − R 4 ( x 4 , x 5 , x 6 , x 7 ) 0 4 n I n ⊗ L T 4 ( x 0 , x 1 , x 2 , x 3 ) 1 C C A (48) l ≥ 1 . So 8 divides i ⊕ j as 8 divides both (16 n )(1 − 2 − l ) an d 8 l m . So i %8 = j %8 i.e., i ⊕ j = (( i ⊕ i %8) ⊕ ( j ⊕ j %8)) . Thus it is enou gh to prove th at | ( i %8) · ψ 16 n ( i ⊕ j ) | ≡ 0 . I t is indeed true as ψ 16 n ( i ⊕ j ) is a multiple of 8 . T o prove 4 ), we hav e to show that | (8 i ) · ψ 16 n (8 n ⊕ 8 i ⊕ 8 j ) | ≡ | ( i · ψ n (( i ⊕ j ) . W e hav e 8 n ⊕ 8 i ⊕ 8 j = (1 6 n )(1 − 2 − l ) + 8 l m for some l with l ≥ 1 and m ∈ Z 8 . Let 16 n = 2 a and a = 4 c + d . If l = c , we have ψ 16 n (8 n ⊕ 8 i ⊕ 8 j ) = 8 l χ 2 d ( m ) and ψ n ( i ⊕ j ) = 8 l − 1 χ 2 d ( m ) . On e can easily see that the a bove statement holds. On the othe r han d, if l < c , we have ψ 16 n (8 n ⊕ 8 i ⊕ 8 j ) = (16 n )2 − l − 1 + 8 l χ 8 ( m ) and ψ n ( i ⊕ j ) = n. 2 − l + 8 l − 1 χ 8 ( m ) . In this case too, the statement holds. T o prove 5 ), we hav e to show that | ( i ⊕ 8 n ) · ψ 16 n ( i ⊕ j ) | ≡ | i · ψ 16 n ( i ⊕ j ) | , i.e., | (8 n ) · ψ 16 n ( i ⊕ j ) | ≡ 0 . Now for i ⊕ j = 0 o r g reater than 8 n , (8 n ) · ψ 16 n ( i ⊕ j ) = 0 . T o prove 6 ), we hav e to show that | ( i ⊕ 8 n ) · ψ 16 n ( i ⊕ j ) | ≡ 1 + | i · ψ 16 n ( i ⊕ j ) | , i.e., | (8 n ) · ψ 16 n ( i ⊕ j ) | ≡ 1 . But (8 n ) · ψ 16 n ( i ⊕ j ) = 8 n for all ( i ⊕ j ) ∈ { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 n } . B. Adams-Lax-P hillips Construction fr om Quaternio ns a nd Geramita-Pullman Construction as special cases Adams-Lax- Phillips h as also provid ed ano ther co nstruction of square R ODs using Quaternio ns wh ich is explicitly shown in [1]. Assuming that a square R OD of or der n = 2 a O ( Q ) n = O ( Q ) n ( x 0 , · · · , x ρ ( n ) − 1 ) which has ρ ( n ) real variables, is given, then a square R O D of ord er 1 6 n with ρ ( n ) + 8 real variables x i for i = 0 , 1 , · · · , ρ ( n ) + 7 O ( Q ) 16 n = O ( Q ) 16 n ( x 0 , · · · , x ρ ( n )+7 ) is given by (48), as shown at the top of this p age wh ere the two matrices L 4 and R 4 are given by L 4 ( x 0 , x 1 , x 2 , x 3 ) =     x 0 x 1 x 2 x 3 − x 1 x 0 − x 3 x 2 − x 2 x 3 x 0 − x 1 − x 3 − x 2 x 1 x 0     , R 4 ( x 4 , x 5 , x 6 , x 7 ) =     x 4 x 5 x 6 x 7 − x 5 x 4 x 7 − x 6 − x 6 − x 7 x 4 x 5 − x 7 x 6 − x 5 x 4     . respectively with y i = x 8+ i . The Geramita-Pullm an construction o f sq uare ROD is also giv en by induction explicitly in [1]. Consider a recu rsiv e co nstruction o f square ROD of order n = 2 a to 16 n as follows: O ( GP ) n = O ( GP ) n ( x 0 , · · · , x ρ ( n ) − 1 ) which has ρ ( n ) real variables is g i ven, then a sq uare R OD O ( GP ) 16 n of order 16 n with ρ ( n ) + 8 real variables x i for i = 0 , 1 , · · · , ρ ( n ) + 7 is given by " K 8 ( x 0 , · · · , x 7 ) ⊗ I n I 8 ⊗ O ( GP ) n ( y 0 , · · · , y ρ ( n ) − 1 ) I 8 ⊗ ( − O ( GP ) n ) T ( y 0 , · · · , y ρ ( n ) − 1 ) K T 8 ( x 0 , · · · , x 7 ) ⊗ I n # (51) with y i = x 8+ i . It can be checked that bo th Adams-Lax-Ph illips construction from Quaternion s and Geram ita-Pullman’ s co nstruction giv e n in [1] differ from the constructions of O ( Q ) 16 n and O ( GP ) 16 n deﬁned above only in rear rangeme nt of variables an d in signs of some of the rows o r columns of the design matrix. Lemma 7: L et t ≥ 16 and ˆ A t and P t be the square ROD o f order t as giv e n in Le mma 5 (ii) and (iii), and also let O ( Q ) 16 n and O ( GP ) 16 n be the squa re R ODs giv en in (48) and in (51) which ar e of ord er 16 n . Then ˆ A t = O ( Q ) 16 n and P t = O ( GP ) 16 n for t = 1 6 n. Pr oof: Similar to that of Lemma 6 and hence omitted. Example 5: Sq uare R OD A 16 of size [16 , 16 , 9] by Adam s- Lax-Phillips construction from Octonion is given by (52). Square R OD ˆ A 16 of size [16 , 16 , 9] by Adams-Lax-Phillip s construction fro m Quatern ion is given by ( 53). Squar e R OD of P 16 size [1 6 , 16 , 9] by Geramita-Pullman co nstruction is th e same as R 16 . Square R OD of P 32 size [32 , 32 , 10] by Geramita-Pu llman construction is given by (54). 16                             x 0 x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 0 0 0 0 0 0 0 − x 1 x 0 − x 3 x 2 − x 5 x 4 x 7 − x 6 0 x 8 0 0 0 0 0 0 − x 2 x 3 x 0 − x 1 − x 6 − x 7 x 4 x 5 0 0 x 8 0 0 0 0 0 − x 3 − x 2 x 1 x 0 − x 7 x 6 − x 5 x 4 0 0 0 x 8 0 0 0 0 − x 4 x 5 x 6 x 7 x 0 − x 1 − x 2 − x 3 0 0 0 0 x 8 0 0 0 − x 5 − x 4 x 7 − x 6 x 1 x 0 x 3 − x 2 0 0 0 0 0 x 8 0 0 − x 6 − x 7 − x 4 x 5 x 2 − x 3 x 0 x 1 0 0 0 0 0 0 x 8 0 − x 7 x 6 − x 5 − x 4 x 3 x 2 − x 1 x 0 0 0 0 0 0 0 0 x 8 x 8 0 0 0 0 0 0 0 − x 0 x 1 x 2 x 3 x 4 x 5 x 6 x 7 0 x 8 0 0 0 0 0 0 − x 1 − x 0 − x 3 x 2 − x 5 x 4 x 7 − x 6 0 0 x 8 0 0 0 0 0 − x 2 x 3 − x 0 − x 1 − x 6 − x 7 x 4 x 5 0 0 0 x 8 0 0 0 0 − x 3 − x 2 x 1 − x 0 − x 7 x 6 − x 5 x 4 0 0 0 0 x 8 0 0 0 − x 4 x 5 x 6 x 7 − x 0 − x 1 − x 2 − x 3 0 0 0 0 0 x 8 0 0 − x 5 − x 4 x 7 − x 6 x 1 − x 0 x 3 − x 2 0 0 0 0 0 0 x 8 0 − x 6 − x 7 − x 4 x 5 x 2 − x 3 − x 0 x 1 0 0 0 0 0 0 0 x 8 − x 7 x 6 − x 5 − x 4 x 3 x 2 − x 1 − x 0                             (52)                             x 0 x 1 x 2 x 3 0 0 0 0 x 4 x 5 x 6 x 7 x 8 0 0 0 − x 1 x 0 − x 3 x 2 0 0 0 0 − x 5 x 4 − x 7 − x 6 0 x 8 0 0 − x 2 x 3 x 0 − x 1 0 0 0 0 − x 6 − x 7 x 4 x 5 0 0 x 8 0 − x 3 − x 2 x 1 x 0 0 0 0 0 − x 7 x 6 − x 5 x 4 0 0 0 x 8 0 0 0 0 x 0 x 1 x 2 x 3 − x 8 0 0 0 x 4 − x 5 − x 6 − x 7 0 0 0 0 − x 1 x 0 − x 3 x 2 0 − x 8 0 0 x 5 x 4 − x 7 x 6 0 0 0 0 − x 2 x 3 x 0 − x 1 0 0 − x 8 0 x 6 − x 7 x 4 − x 5 0 0 0 0 − x 3 − x 2 x 1 x 0 0 0 0 − x 8 x 7 − x 6 x 5 x 4 − x 4 x 5 x 6 x 7 x 8 0 0 0 x 0 − x 1 − x 2 − x 3 0 0 0 0 − x 5 − x 4 x 7 − x 6 0 x 8 0 0 x 1 x 0 x 3 − x 2 0 0 0 0 − x 6 x 7 − x 4 x 5 0 0 x 8 0 x 2 − x 3 x 0 x 1 0 0 0 0 − x 7 x 6 − x 5 − x 4 0 0 0 x 8 x 3 x 2 − x 1 x 0 0 0 0 0 − x 8 0 0 0 − x 4 − x 5 − x 6 − x 7 0 0 0 0 x 0 − x 1 − x 2 − x 3 0 − x 8 0 0 x 5 − x 4 x 7 x 6 0 0 0 0 x 1 x 0 x 3 − x 2 0 0 − x 8 0 x 6 x 7 − x 4 − x 5 0 0 0 0 x 2 − x 3 x 0 x 1 0 0 0 − x 8 x 7 − x 6 x 5 − x 4 0 0 0 0 x 3 x 2 − x 1 x 0                             (53) 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 x 0 0 x 1 0 x 2 0 x 3 0 x 4 0 x 5 0 x 6 0 x 7 0 x 8 x 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x 0 0 x 1 0 x 2 0 x 3 0 x 4 0 x 5 0 x 6 0 x 7 − x 9 x 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 − x 1 0 x 0 0 − x 3 0 x 2 0 − x 5 0 x 4 0 x 7 0 − x 6 0 0 0 x 8 x 9 0 0 0 0 0 0 0 0 0 0 0 0 0 − x 1 0 x 0 0 − x 3 0 x 2 0 − x 5 0 x 4 0 x 7 0 − x 6 0 0 − x 9 x 8 0 0 0 0 0 0 0 0 0 0 0 0 − x 2 0 x 3 0 x 0 0 − x 1 0 − x 6 0 − x 7 0 x 4 0 x 5 0 0 0 0 0 x 8 x 9 0 0 0 0 0 0 0 0 0 0 0 − x 2 0 x 3 0 x 0 0 − x 1 0 − x 6 0 − x 7 0 x 4 0 x 5 0 0 0 0 − x 9 x 8 0 0 0 0 0 0 0 0 0 0 − x 3 0 − x 2 0 x 1 0 x 0 0 − x 7 0 x 6 0 − x 5 0 x 4 0 0 0 0 0 0 0 x 8 x 9 0 0 0 0 0 0 0 0 0 − x 3 0 − x 2 0 x 1 0 x 0 0 − x 7 0 x 6 0 − x 5 0 x 4 0 0 0 0 0 0 − x 9 x 8 0 0 0 0 0 0 0 0 − x 4 0 x 5 0 x 6 0 x 7 0 x 0 0 − x 1 0 − x 2 0 − x 3 0 0 0 0 0 0 0 0 0 x 8 x 9 0 0 0 0 0 0 0 − x 4 0 x 5 0 x 6 0 x 7 0 x 0 0 − x 1 0 − x 2 0 − x 3 0 0 0 0 0 0 0 0 − x 9 x 8 0 0 0 0 0 0 − x 5 0 − x 4 0 x 7 0 − x 6 0 x 1 0 x 0 0 x 3 0 − x 2 0 0 0 0 0 0 0 0 0 0 0 x 8 x 9 0 0 0 0 0 − x 5 0 − x 4 0 x 7 0 − x 6 0 x 1 0 x 0 0 x 3 0 − x 2 0 0 0 0 0 0 0 0 0 0 − x 9 x 8 0 0 0 0 − x 6 0 − x 7 0 − x 4 0 x 5 0 x 2 0 − x 3 0 x 0 0 x 1 0 0 0 0 0 0 0 0 0 0 0 0 0 x 8 x 9 0 0 0 − x 6 0 − x 7 0 − x 4 0 x 5 0 x 2 0 − x 3 0 x 0 0 x 1 0 0 0 0 0 0 0 0 0 0 0 0 − x 9 x 8 0 0 − x 7 0 x 6 0 − x 5 0 − x 4 0 x 3 0 x 2 0 − x 1 0 x 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x 8 x 9 0 − x 7 0 x 6 0 − x 5 0 − x 4 0 x 3 0 x 2 0 − x 1 0 x 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 − x 9 x 8 − x 8 x 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x 0 0 − x 1 0 − x 2 0 − x 3 0 − x 4 0 − x 5 0 − x 6 0 − x 7 0 − x 9 − x 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x 0 0 − x 1 0 − x 2 0 − x 3 0 − x 4 0 − x 5 0 − x 6 0 − x 7 0 0 − x 8 x 9 0 0 0 0 0 0 0 0 0 0 0 0 x 1 0 x 0 0 x 3 0 − x 2 0 x 5 0 − x 4 0 − x 7 0 x 6 0 0 0 − x 9 − x 8 0 0 0 0 0 0 0 0 0 0 0 0 0 x 1 0 x 0 0 x 3 0 − x 2 0 x 5 0 − x 4 0 − x 7 0 x 6 0 0 0 0 − x 8 x 9 0 0 0 0 0 0 0 0 0 0 x 2 0 − x 3 0 x 0 0 x 1 0 x 6 0 x 7 0 − x 4 0 − x 5 0 0 0 0 0 − x 9 − x 8 0 0 0 0 0 0 0 0 0 0 0 x 2 0 − x 3 0 x 0 0 x 1 0 x 6 0 x 7 0 − x 4 0 − x 5 0 0 0 0 0 0 − x 8 x 9 0 0 0 0 0 0 0 0 x 3 0 x 2 0 − x 1 0 x 0 0 x 7 0 − x 6 0 x 5 0 − x 4 0 0 0 0 0 0 0 − x 9 − x 8 0 0 0 0 0 0 0 0 0 x 3 0 x 2 0 − x 1 0 x 0 0 x 7 0 − x 6 0 x 5 0 − x 4 0 0 0 0 0 0 0 0 − x 8 x 9 0 0 0 0 0 0 x 4 0 − x 5 0 − x 6 0 − x 7 0 x 0 0 x 1 0 x 2 0 x 3 0 0 0 0 0 0 0 0 0 − x 9 − x 8 0 0 0 0 0 0 0 x 4 0 − x 5 0 − x 6 0 − x 7 0 x 0 0 x 1 0 x 2 0 x 3 0 0 0 0 0 0 0 0 0 0 − x 8 x 9 0 0 0 0 x 5 0 x 4 0 − x 7 0 x 6 0 − x 1 0 x 0 0 − x 3 0 x 2 0 0 0 0 0 0 0 0 0 0 0 − x 9 − x 8 0 0 0 0 0 x 5 0 x 4 0 − x 7 0 x 6 0 − x 1 0 x 0 0 − x 3 0 x 2 0 0 0 0 0 0 0 0 0 0 0 0 − x 8 x 9 0 0 x 6 0 x 7 0 x 4 0 − x 5 0 − x 2 0 x 3 0 x 0 0 − x 1 0 0 0 0 0 0 0 0 0 0 0 0 0 − x 9 − x 8 0 0 0 x 6 0 x 7 0 x 4 0 − x 5 0 − x 2 0 x 3 0 x 0 0 − x 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 − x 8 x 9 x 7 0 − x 6 0 x 5 0 x 4 0 − x 3 0 − x 2 0 x 1 0 x 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 − x 9 − x 8 0 x 7 0 − x 6 0 x 5 0 x 4 0 − x 3 0 − x 2 0 x 1 0 x 0 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 (54) 17 A P P E N D I X C R AT E 1 / 2 S C A L E D C O D O F S I Z E [64 , 1 0 , 32] 1 √ 2 · 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 x 0 − x 1 − x 2 − x 3 − x 4 − x 5 − x 6 − x 7 − x 8 − x 16 x 1 x 0 x 3 − x 2 x 5 − x 4 − x 7 x 6 x 9 x 17 x 2 − x 3 x 0 x 1 x 6 x 7 − x 4 − x 5 x 10 x 18 x 3 x 2 − x 1 x 0 x 7 − x 6 x 5 − x 4 x 11 x 19 x 4 − x 5 − x 6 − x 7 x 0 x 1 x 2 x 3 x 12 x 20 x 5 x 4 − x 7 x 6 − x 1 x 0 − x 3 x 2 x 13 x 21 x 6 x 7 x 4 − x 5 − x 2 x 3 x 0 − x 1 x 14 x 22 x 7 − x 6 x 5 x 4 − x 3 − x 2 x 1 x 0 x 15 x 23 x 8 − x 9 − x 10 − x 11 − x 12 − x 13 − x 14 − x 15 x 0 x 24 x 9 x 8 − x 11 x 10 − x 13 x 12 x 15 − x 14 − x 1 x 25 x 10 x 11 x 8 − x 9 − x 14 − x 15 x 12 x 13 − x 2 x 26 x 11 − x 10 x 9 x 8 − x 15 x 14 − x 13 x 12 − x 3 x 27 x 12 x 13 x 14 x 15 x 8 − x 9 − x 10 − x 11 − x 4 x 28 x 13 − x 12 x 15 − x 14 x 9 x 8 x 11 − x 10 − x 5 x 29 x 14 − x 15 − x 12 x 13 x 10 − x 11 x 8 x 9 − x 6 x 30 x 15 x 14 − x 13 − x 12 x 11 x 10 − x 9 x 8 − x 7 x 31 x 16 − x 17 − x 18 − x 19 − x 20 − x 21 − x 22 − x 23 − x 24 x 0 x 17 x 16 x 19 − x 18 x 21 − x 20 − x 23 x 22 x 25 − x 1 x 18 − x 19 x 16 x 17 x 22 x 23 − x 20 − x 21 x 26 − x 2 x 19 x 18 − x 17 x 16 x 23 − x 22 x 21 − x 20 x 27 − x 3 x 20 − x 21 − x 22 − x 23 x 16 x 17 x 18 x 19 x 28 − x 4 x 21 x 20 − x 23 x 22 − x 17 x 16 − x 19 x 18 x 29 − x 5 x 22 x 23 x 20 − x 21 − x 18 x 19 x 16 − x 17 x 30 − x 6 x 23 − x 22 x 21 x 20 − x 19 − x 18 x 17 x 16 x 31 − x 7 x 24 − x 25 − x 26 − x 27 − x 28 − x 29 − x 30 − x 31 x 16 − x 8 x 25 x 24 − x 27 x 26 − x 29 x 28 x 31 − x 30 − x 17 − x 9 x 26 x 27 x 24 − x 25 − x 30 − x 31 x 28 x 29 − x 18 − x 10 x 27 − x 26 x 25 x 24 − x 31 x 30 − x 29 x 28 − x 19 − x 11 x 28 x 29 x 30 x 31 x 24 − x 25 − x 26 − x 27 − x 20 − x 12 x 29 − x 28 x 31 − x 30 x 25 x 24 x 27 − x 26 − x 21 − x 13 x 30 − x 31 − x 28 x 29 x 26 − x 27 x 24 x 25 − x 22 − x 14 x 31 x 30 − x 29 − x 28 x 27 x 26 − x 25 x 24 − x 23 − x 15 x ∗ 0 − x ∗ 1 − x ∗ 2 − x ∗ 3 − x ∗ 4 − x ∗ 5 − x ∗ 6 − x ∗ 7 − x ∗ 8 − x ∗ 16 x ∗ 1 x ∗ 0 x ∗ 3 − x ∗ 2 x ∗ 5 − x ∗ 4 − x ∗ 7 x ∗ 6 x ∗ 9 x ∗ 17 x ∗ 2 − x ∗ 3 x ∗ 0 x ∗ 1 x ∗ 6 x ∗ 7 − x ∗ 4 − x ∗ 5 x ∗ 10 x ∗ 18 x ∗ 3 x ∗ 2 − x ∗ 1 x ∗ 0 x ∗ 7 − x ∗ 6 x ∗ 5 − x ∗ 4 x ∗ 11 x ∗ 19 x ∗ 4 − x ∗ 5 − x ∗ 6 − x ∗ 7 x ∗ 0 x ∗ 1 x ∗ 2 x ∗ 3 x ∗ 12 x ∗ 20 x ∗ 5 x ∗ 4 − x ∗ 7 x ∗ 6 − x ∗ 1 x ∗ 0 − x ∗ 3 x ∗ 2 x ∗ 13 x ∗ 21 x ∗ 6 x ∗ 7 x ∗ 4 − x ∗ 5 − x ∗ 2 x ∗ 3 x ∗ 0 − x ∗ 1 x ∗ 14 x ∗ 22 x ∗ 7 − x ∗ 6 x ∗ 5 x ∗ 4 − x ∗ 3 − x ∗ 2 x ∗ 1 x ∗ 0 x ∗ 15 x ∗ 23 x ∗ 8 − x ∗ 9 − x ∗ 10 − x ∗ 11 − x ∗ 12 − x ∗ 13 − x ∗ 14 − x ∗ 15 x ∗ 0 x ∗ 24 x ∗ 9 x ∗ 8 − x ∗ 11 x ∗ 10 − x ∗ 13 x ∗ 12 x ∗ 15 − x ∗ 14 − x ∗ 1 x ∗ 25 x ∗ 10 x ∗ 11 x ∗ 8 − x ∗ 9 − x ∗ 14 − x ∗ 15 x ∗ 12 x ∗ 13 − x ∗ 2 x ∗ 26 x ∗ 11 − x ∗ 10 x ∗ 9 x ∗ 8 − x ∗ 15 x ∗ 14 − x ∗ 13 x ∗ 12 − x ∗ 3 x ∗ 27 x ∗ 12 x ∗ 13 x ∗ 14 x ∗ 15 x ∗ 8 − x ∗ 9 − x ∗ 10 − x ∗ 11 − x ∗ 4 x ∗ 28 x ∗ 13 − x ∗ 12 x ∗ 15 − x ∗ 14 x ∗ 9 x ∗ 8 x ∗ 11 − x ∗ 10 − x ∗ 5 x ∗ 29 x ∗ 14 − x ∗ 15 − x ∗ 12 x ∗ 13 x ∗ 10 − x ∗ 11 x ∗ 8 x ∗ 9 − x ∗ 6 x ∗ 30 x ∗ 15 x ∗ 14 − x ∗ 13 − x ∗ 12 x ∗ 11 x ∗ 10 − x ∗ 9 x ∗ 8 − x ∗ 7 x ∗ 31 x ∗ 16 − x ∗ 17 − x ∗ 18 − x ∗ 19 − x ∗ 20 − x ∗ 21 − x ∗ 22 − x ∗ 23 − x ∗ 24 x ∗ 0 x ∗ 17 x ∗ 16 x ∗ 19 − x ∗ 18 x ∗ 21 − x ∗ 20 − x ∗ 23 x ∗ 22 x ∗ 25 − x ∗ 1 x ∗ 18 − x ∗ 19 x ∗ 16 x ∗ 17 x ∗ 22 x ∗ 23 − x ∗ 20 − x ∗ 21 x ∗ 26 − x ∗ 2 x ∗ 19 x ∗ 18 − x ∗ 17 x ∗ 16 x ∗ 23 − x ∗ 22 x ∗ 21 − x ∗ 20 x ∗ 27 − x ∗ 3 x ∗ 20 − x ∗ 21 − x ∗ 22 − x ∗ 23 x ∗ 16 x ∗ 17 x ∗ 18 x ∗ 19 x ∗ 28 − x ∗ 4 x ∗ 21 x ∗ 20 − x ∗ 23 x ∗ 22 − x ∗ 17 x ∗ 16 − x ∗ 19 x ∗ 18 x ∗ 29 − x ∗ 5 x ∗ 22 x ∗ 23 x ∗ 20 − x ∗ 21 − x ∗ 18 x ∗ 19 x ∗ 16 − x ∗ 17 x ∗ 30 − x ∗ 6 x ∗ 23 − x ∗ 22 x ∗ 21 x ∗ 20 − x ∗ 19 − x ∗ 18 x ∗ 17 x ∗ 16 x ∗ 31 − x ∗ 7 x ∗ 24 − x ∗ 25 − x ∗ 26 − x ∗ 27 − x ∗ 28 − x ∗ 29 − x ∗ 30 − x ∗ 31 x ∗ 16 − x ∗ 8 x ∗ 25 x ∗ 24 − x ∗ 27 x ∗ 26 − x ∗ 29 x ∗ 28 x ∗ 31 − x ∗ 30 − x ∗ 17 − x ∗ 9 x ∗ 26 x ∗ 27 x ∗ 24 − x ∗ 25 − x ∗ 30 − x ∗ 31 x ∗ 28 x ∗ 29 − x ∗ 18 − x ∗ 10 x ∗ 27 − x ∗ 26 x ∗ 25 x ∗ 24 − x ∗ 31 x ∗ 30 − x ∗ 29 x ∗ 28 − x ∗ 19 − x ∗ 11 x ∗ 28 x ∗ 29 x ∗ 30 x ∗ 31 x ∗ 24 − x ∗ 25 − x ∗ 26 − x ∗ 27 − x ∗ 20 − x ∗ 12 x ∗ 29 − x ∗ 28 x ∗ 31 − x ∗ 30 x ∗ 25 x ∗ 24 x ∗ 27 − x ∗ 26 − x ∗ 21 − x ∗ 13 x ∗ 30 − x ∗ 31 − x ∗ 28 x ∗ 29 x ∗ 26 − x ∗ 27 x ∗ 24 x ∗ 25 − x ∗ 22 − x ∗ 14 x ∗ 31 x ∗ 30 − x ∗ 29 − x ∗ 28 x ∗ 27 x ∗ 26 − x ∗ 25 x ∗ 24 − x ∗ 23 − x ∗ 15 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 .

Low-delay, Low-PAPR, High-rate Non-square Complex Orthogonal Designs

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