Square Complex Orthogonal Designs with Low PAPR and Signaling Complexity

IEEE TRANSACTIONS ON WIRELESS COMMUNICA TIONS , V OL. XX, NO. XX, XXXX 1 Square Comple x Orthogonal Designs with Lo w P APR and Signaling Comple xity Smarajit Das, Student Membe r , IEEE an d B. Sundar Rajan, Senior Member , IEEE Abstract —Space-Time Block Codes from square complex or - thogonal d esigns ( SCOD) have b een extensively stud ied and most of the existing SCODs contain larg e n umber of zero. The zeros in the d esigns result in high peak-to-a verage po wer ratio (P APR) and also impose a sev ere constraint on h ardwar e impl ementation of the code when turning off some of t he transmitting antennas whenever a zero is transmitted. Recently , rate 1 2 SCODs with no zero entry hav e b een reported for 8 transmit antennas. In this paper , S CODs with no zero entry for 2 a transmit antennas whenever a + 1 is a power of 2 , are constructed which in cludes the 8 transmit antennas case as a special case. More generally , fo r arbitrary v alues of a , explicit construction of 2 a × 2 a rate a +1 2 a SCODs with the ratio of number of zero entries to the total number of entries equal to 1 − a +1 2 a 2 ⌊ log 2 ( 2 a a +1 ) ⌋ is reported, wher eas fo r standard known constructions, the ratio is 1 − a +1 2 a . The codes presented d o not result in in crea sed signaling complexity . Simulation results show th at the codes constructed in this paper outperf orm the codes using the standard construction under peak p ower constraint while perform ing the same un der a verage power constraint. Index T erms —Amicable orthogonal designs, M IMO, orthogo- nal d esigns, P A PR, space-time codes, transmit d iv ersity . I . I N T RO D U C T I O N S P ACE-TIME Block Codes (STBCs) from Complex Or- thogon al Designs (CODs) have been extensi vely studied in [1], [2], [3]. Let x 1 , x 2 , · · · , x t be com muting, real indeterminates. A real o rthogo nal design X of order n and typ e ( a 1 , a 2 , · · · , a t ) , denoted as OD ( n ; a 1 , a 2 , · · · , a t ) where the coefficients a i are positive integers, is a matrix o f order n with entries chosen from 0 , ± x 1 , ± x 2 , · · · , ± x t , such that X T X = ( a 1 x 2 1 + a 2 x 2 2 + · · · + a t x 2 t ) I n where X T denotes the transpo se of the matrix X an d I n is the n × n iden tity matrix . Amicable ortho gonal design s (A ODs) are defined using two real or thogon al designs of same o rder but not neces- sarily of same type. Let X be an O D ( n ; u 1 , u 2 , · · · , u s ) on the re al variables x 1 , x 2 , · · · , x s and let Y b e an OD ( n ; v 1 , v 2 , · · · , v t ) o n the real variables y 1 , y 2 , · · · , y t . It is said that X and Y are AOD ( n ; u 1 , u 2 , · · · , u s ; v 1 , v 2 , · · · , v t ) if XY T = YX T . This work was supporte d through grants to B.S. Rajan; partly by the IISc-DRDO program on Advance d Research in Mathematica l Engineeri ng, and partly by the Council of Scientific & Industrial Research (CSIR, India) Research Grant (22(0365)/04/EMR-II). The material in this paper was presente d in parts at the IEEE International Symposium on Informa- tion theory held at Nice, France during June 24-29, 2007. Smarajit Das and B. Sundar Rajan are with t he Depa rtment of Elec trical Communi- catio n Engineeri ng, Indian Institute of Science, Bangalore- 560012, India. Email: { smaraj it,bsrajan } @ece.iisc.ernet.in. Manuscript rece iv ed July 15, 2007; re vised No vember 22, 2007. Amicable or thogon al designs h a ve be en studied b y several authors [10], [11] to construct comp lex o rthogon al design s. The b ook by Geramita an d Seberry [11] gives a nic e intro - duction to this topic. In the following, we define square co mplex orthog onal de- sign which we use freq uently in the rest of the paper . A Square Complex Orthogona l Design ( SCOD) G ( x 1 , x 2 , ..., x k ) (in short G ) of size n is an n × n matrix such that: • the entries of G ( x 1 , x 2 , ..., x k ) are co mplex linear com- binations o f the variables x 1 , x 2 , ..., x k and their complex conjuga tes x ∗ 1 , x ∗ 2 , ..., x ∗ k , • G H G = ( | x 1 | 2 + ... + | x k | 2 ) I n where H stan ds for the complex conjugate transpo se and I n is the n × n identity matrix. If the non-zero entries are the in determinates ± x 1 , · · · , ± x k or their conjugates ± x ∗ 1 , ± x ∗ 2 , ..., ± x ∗ k only (no t arbitra ry complex linear co mbinations), then G is said to be a r estr icted complex orthogona l design (RCOD). The rate of G is k n complex sym bols per chan nel use. It is k nown that the maximum rate R of an n × n RCOD is a +1 n where n = 2 a (2 b + 1) , a and b are positiv e integers [2]. Note that the maxima l rate d oes not dep end on b. Several authors hav e constructed RCODs for 2 a antennas achieving maximal r ate [ 2], [4], [5], [6]. In [ 2], the fo llowing indu ction method is used to construct SCODs fo r 2 a antennas, a = 2 , 3 , · · · , starting from G 1 =  x 1 − x ∗ 2 x 2 x ∗ 1  , G a =  G a − 1 − x ∗ a +1 I 2 a − 1 x a +1 I 2 a − 1 G H a − 1  (1) where G a is a 2 a × 2 a complex matrix. No te that G a is a RCOD in a + 1 comp lex variables x 1 , x 2 , · · · , x a +1 . Moreover , each r ow and each column o f the matrix G a contains only a + 1 n on-zero elements an d a ll oth er e ntries in th e sam e row or co lumn are filled with zeros. The fraction o f zeros, de fined as th e ratio of the n umber of zeros to the total n umber of entries in a design , for G a , is 2 a − a − 1 2 a = 1 − a + 1 2 a = 1 − R . (2) For the co nstructions in [2], [4], [5], [6] also, th e fractio n of zeros is the same as giv en by (2). Redu cing number of zer os in a SCOD for mor e than 2 tran smit antenn as (for two antenn as, the Alamouti code does not have any zeros), is impo rtant for many reasons, namely improvement in Peak-to -A verage P ower Ratio (P APR) and also the ease o f practical implementation o f these c odes in wireless commu nication system [ 13]. For illustration, con sider the SCOD G 2 of size 4 shown below - it is a RCOD, whereas the code G T J C also shown 2 IEEE TRANSACTIONS ON WIREL ESS COMMUNICA TIONS , VOL. XX, NO. XX , XXXX G T W M S = 1 √ 2 2 6 6 6 6 6 6 6 6 6 6 6 4 x 1 x 1 x 2 x 2 x 3 x 4 x 3 x 4 x 1 − x 1 x 2 − x 2 x ∗ 4 − x ∗ 3 x ∗ 4 − x ∗ 3 x ∗ 2 x ∗ 2 − x ∗ 1 − x ∗ 1 x 3 x 4 − x 3 − x 4 x ∗ 2 − x ∗ 2 − x ∗ 1 x ∗ 1 x ∗ 4 − x ∗ 3 − x ∗ 4 x ∗ 3 x 4 I + jx 3 Q x 3 I + jx 4 Q x 4 I + jx 3 Q x 3 I + jx 4 Q x 2 I + jx 1 Q x 2 I + jx 1 Q x 1 I + jx 2 Q x 1 I + jx 2 Q x 3 I + jx 4 Q x 4 I + jx 3 Q x 3 I + jx 4 Q x 4 I + jx 3 Q x 2 I + jx 1 Q x 2 I + jx 1 Q x 1 I + jx 2 Q x 1 I + jx 2 Q x 4 I + jx 3 Q x 3 I + jx 4 Q x 4 I + jx 3 Q x 3 I + jx 4 Q x 1 I + jx 2 Q x 1 I + jx 2 Q x 2 I + jx 1 Q x 2 I + jx 1 Q x 3 I + jx 4 Q x 4 I + jx 3 Q x 3 I + jx 4 Q x 4 I + jx 3 Q x 1 I + jx 2 Q x 1 I + jx 2 Q x 2 I + jx 1 Q x 2 I + jx 1 Q 3 7 7 7 7 7 7 7 7 7 7 7 5 below , given in [1], [11], obtained from Amicab le Ortho gonal Designs, is not a RCOD and th ere are n o zeros in this matrix . G 2 = 2 6 4 x 1 − x ∗ 2 − x ∗ 3 0 x 2 x ∗ 1 0 − x ∗ 3 x 3 0 x ∗ 1 x ∗ 2 0 x 3 − x 2 x 1 3 7 5 , G T J C = 2 6 6 6 4 x 1 x 2 x 3 √ 2 x 3 √ 2 − x ∗ 2 x ∗ 1 x 3 √ 2 − x 3 √ 2 x ∗ 3 √ 2 x ∗ 3 √ 2 ( − x 1 − x ∗ 1 + x 2 − x ∗ 2 ) 2 ( x 1 − x ∗ 1 − x 2 − x ∗ 2 ) 2 x ∗ 3 √ 2 − x ∗ 3 √ 2 ( x 1 − x ∗ 1 + x 2 + x ∗ 2 ) 2 − ( x 1 + x ∗ 1 + x 2 − x ∗ 2 ) 2 3 7 7 7 5 (3) Notice that some of th e entr ies of G T J C can be written as ( − x 1 − x ∗ 1 + x 2 − x ∗ 2 ) 2 = − ( x 1 I − j x 2 Q ) = − ˆ x ∗ 1 , ( x 1 − x ∗ 1 − x 2 − x ∗ 2 ) 2 = − ( x 2 I − j x 1 Q ) = − ˆ x ∗ 2 , ( x 1 − x ∗ 1 + x 2 + x ∗ 2 ) 2 = x 2 I + j x 1 Q = ˆ x 2 , − ( x 1 + x ∗ 1 + x 2 − x ∗ 2 ) 2 = − ( x 1 I + j x 2 Q ) = − ˆ x 1 , (4) where ˆ x 1 = x 1 I + j x 2 Q and ˆ x 2 = x 2 I + j x 1 Q are the coordin ate in terleaved variables cor respondin g to the vari- ables x 1 and x 2 , where x iI and x iQ are the in- phase and the quad rature-p hase of the variable x i . Sing le-Symbol ML Decodable Designs based on co ordinate inter lea ved variables have been studied in [ 12]. For ou r purp oses, it is impo rtant to note that wh enever co ordinate interleaving app ears, it is nothing but a specific complex linear combin ation of two variables, wh ich will have impact in terms of the signaling complexity explained subseque ntly . The following code G 3 for 8 transm it anten nas, G 3 =             x 1 − x ∗ 2 − x ∗ 3 0 − x ∗ 4 0 0 0 x 2 x ∗ 1 0 − x ∗ 3 0 − x ∗ 4 0 0 x 3 0 x ∗ 1 x ∗ 2 0 0 − x ∗ 4 0 0 x 3 − x 2 x 1 0 0 0 − x ∗ 4 x 4 0 0 0 x ∗ 1 x ∗ 2 x ∗ 3 0 0 x 4 0 0 − x 2 x 1 0 x ∗ 3 0 0 x 4 0 − x 3 0 x 1 − x ∗ 2 0 0 0 x 4 0 − x 3 x 2 x ∗ 1             , G Y =             x ∗ 1 x ∗ 1 x 2 − x 2 x 3 − x 3 x 4 − x 4 j x 1 − j x 1 j x ∗ 2 j x ∗ 2 j x ∗ 3 j x ∗ 3 j x ∗ 4 j x ∗ 4 − x 2 x 2 x ∗ 1 x ∗ 1 x ∗ 4 − x ∗ 4 − x ∗ 3 x ∗ 3 − j x ∗ 2 − j x ∗ 2 j x 1 − j x 1 j x 4 j x 4 − j x 3 − j x 3 − x 3 x 3 − x ∗ 4 x ∗ 4 x ∗ 1 x ∗ 1 x ∗ 2 − x ∗ 2 − j x ∗ 3 − j x ∗ 3 − j x 4 − j x 4 j x 1 − j x 1 j x 2 j x 2 − x 4 x 4 x ∗ 3 − x ∗ 3 − x ∗ 2 x ∗ 2 x ∗ 1 x ∗ 1 − j x ∗ 4 − j x ∗ 4 j x 3 j x 3 − j x 2 − j x 2 j x 1 − j x 1             (5) contains 50 p er cent of en tries zeros. But, Y uen et al, in [7], have constru cted a new r ate- 1 / 2 , SCOD G Y √ 2 of size 8 with no zeros in the desig n matr ix usin g Amicab le Comp lex Orthogo nal Design (A COD) [11] where G Y is giv en in (5). Observe that for a fixed av erage p ower per codeword, du e to the p resence of zer os in G 3 , the pe ak p ower tran smission in an antenna using G 3 will be h igher than th at of an ante nna using G Y . Hence, it is clear tha t the P APR fo r the co de G Y is lower th an th at of the code G 3 . Hence, lower the frac tion of zeros in a co de, lower will be the P APR of the co de. In [8], [9], [10], ano ther rate- 1 / 2 , 8 antenn a cod e with no zero entry , denoted by G T W M S shown at th e top of this pag e, has been repor ted. Observe tha t G T W M S has entries that are coo rdinated in - terleaved variables and hen ce has larger signalin g comp lexity . Signalin g c omplexity: Notice that some of the entries, fo r instance x 1 I + j x 2 Q and x 2 I + j x 1 Q , in G T J C and G T W M S are co-ord inate interlea ved versions of th e variables x 1 and x 2 . Supp ose the variables x 1 and x 2 take values fro m a regular (rectang ular) 16 -QAM rotated by an angle θ . Thoug h rotation does not affect the full-diversity of th e code, the coding g ain depend s o n θ an d hence non-zer o value of θ may be desired. Now the antenn a transmitting x 1 chooses o ne o f th e 16 complex n umbers for transmission where as the antenn a transmitting x 1 I + j x 2 Q will be ch oosing on e of 16 × 16 com plex numbers since the co mponen ts x 1 I and x 2 Q take indep endently 16 v alues each. This will increase the number of quantization le vels needed in a dig ital imp lementation fo r sign als tran smitted in this anten na. W e will hencef orth refer to the n umber o f quantization lev els need ed in such a digital implemen tation as “signaling complexity”. Notice that designs which have entries that are linear co mbinations of sev eral variables increase the signaling complexity o f the design. Accord ingly , the signaling complexity of G 2 giv en in (3) is less than that of th e co de on the righ t h and side of (3). Similarly , the sign aling co mplexity of G T W M S is larger th an that o f G Y . Notice that by multip lying the matrix G 3 with a unitary matrix, the resulting matrix will con tinue to be a SCOD with different number o f zeros and it is not difficult to find unitary matrices that will result in a design with no zero en tries. Howe ver , such a d esign is likely to h av e a large signaling complexity which needs to be a voided. Obtaining a un itary matrix which re duces the nu mber of zero entries while not increasing the signaling complexity is a nontr i vial task which is the subject matter of this pap er . In this pape r , we provid e a gene ral pr ocedure to construc t SCODs with fewer num ber of zero s compar ed to k nown construction s for any p ower of two numb er of anten nas (greater than 4), with out incre asing the sign aling comp lexity . Our contributions are summ arized as f ollows: • Maximal-ra te SCODs with no zero en try and minimum signaling co mplexity for 2 a transmit antenna s whenever DAS and RAJ AN: SQUARE COMPLEX OR THOGONAL DESIGNS WITH LOW P APR AND SIGNALING COMPLEXITY 3 a + 1 is a power of 2 , are c onstructed wh ich includes th e 8 transmit antennas ca se as a spec ial case. T his ma tches with the construc tion g i ven in [7] for 8 tran smit antennas and be ats th e cod es in [ 8], [9 ], [ 10] fo r 8 transmit antennas in terms of signalin g comp lexity . • More genera lly , for arbitrar y values of a , explicit con - struction of 2 a × 2 a , rate a +1 2 a SCODs with the ra tio of number of zero entries to the total number of entries eq ual to 1 − a +1 2 a 2 ⌊ log 2 ( 2 a a +1 ) ⌋ is rep orted. Note that when a + 1 is a p ower of two, ou r code s have no zeros. When a + 1 is not a power of two, it is con jectured that SCODs with smaller fraction of zero entr ies with rate a +1 2 a and same signaling complexity do not exist. Our c onstruction giv es fewer number of zero entries compar ed to the well known construction s in [2], [4], [5], [6]. • Our co nstruction is based o n simple premultiplicatio n of the code in (1) by a scaled unitary matrix consisting of only +1 , − 1 or 0 , whe reas the co nstructions in [7], [8], [ 9], [10] depend on the existence and av ailability of A ODs [ 11]. • A gener al pro cedure to obtain the scaled un itary matrix that leads to a SCOD with small number of zer o e ntries is giv en. • It is sho wn that the new codes presented in this paper admit a r ecursive relation similar to that admitted by G a . The remainin g content of the paper is organized as follows: In Section II , we prove the main result of the paper g iv en by Theorem 1. In Section III, we g iv e a p rocedur e to c ompute the pr emultiplying matrix using wh ich we can ge t the SCODs of this p aper straig htaway from the well-kn own constructio n giv en by (1). Th e P APR of the new codes construc ted is discussed in Section IV. Simulation results ar e gi ven in Section V. A brief summ ary an d a con jecture co nstitute Sectio n VI. I I . C O N S T RU C T I O N O F S C O D S W I T H L O W P A P R SCODs given in [ 2] contain a large number of zero s and the f raction o f zeros in the code in creases as the numbe r of transmit antenna incr eases. Note that these codes are RCODs and hence o f least d ecoding com plexity as well as least signaling co mplexity . It is possible to o btain an ortho gonal matrix with fewer zero, if we premultiply and/or post-m ultiply the giv en orthogonal design ma trix by some unitary matrix, b ut the resulting orth ogona l d esign need not be a RCOD. So care must be taken in how we cho ose these p remultiplyin g or post- multiplying matrices such that the co de obtained after applying these matrices, does no t c ontain comp lex linear c ombination of the variables which will increase the sign aling comp lexity . There exists a unitary matrix which wh en p re-multiplies the code G 3 obtain a code wh ich contain s no zero in the matrix and none of the entries of this n e w c ode is a comp lex linear combinatio n of variables a nd th us the signa ling co mplexity is not increased . The un itary matr ix co rrespond ing to G 3 is 1 √ 2 Q (3) where Q (3) is given by the matr ix on th e lef t han d side o f ( 6). Her e − 1 is rep resented by simply the minus sign (throu ghout the paper) an d th e resu lting no zero entry SCOD is H 3 where the matrix √ 2 H 3 is shown on the right h and side of (6).             1 00 0 0 0 0 1 0 10 0 0 0 1 0 0 01 0 0 1 0 0 0 00 1 1 0 0 0 0 00 1 − 0 0 0 0 01 0 0 − 0 0 0 10 0 0 0 − 0 1 00 0 0 0 0 −             ,             x 1 − x ∗ 2 − x ∗ 3 x 4 − x ∗ 4 − x 3 x 2 x ∗ 1 x 2 x ∗ 1 x 4 − x ∗ 3 − x 3 − x ∗ 4 x 1 − x ∗ 2 x 3 x 4 x ∗ 1 x ∗ 2 − x 2 x 1 − x ∗ 4 x ∗ 3 x 4 x 3 − x 2 x 1 x ∗ 1 x ∗ 2 x ∗ 3 − x ∗ 4 − x 4 x 3 − x 2 x 1 − x ∗ 1 − x ∗ 2 − x ∗ 3 − x ∗ 4 x 3 − x 4 x ∗ 1 x ∗ 2 x 2 − x 1 − x ∗ 4 − x ∗ 3 x 2 x ∗ 1 − x 4 − x ∗ 3 x 3 − x ∗ 4 − x 1 x ∗ 2 x 1 − x ∗ 2 − x ∗ 3 − x 4 − x ∗ 4 x 3 − x 2 − x ∗ 1             (6) T ow ards identify ing such pr emultiplying ma trices f or th e general case, we label the rows of G a as R 0 , R 1 , · · · , R 2 a − 1 . The column ind ex also v aries from 0 to 2 a − 1 . Let N ( a ) i be the set o f co lumn ind ices of the non- zero entries of the i - th row R i of th e matrix G a . The fo llowing lemm a descr ibes N ( a ) i for all i = 0 to 2 a − 1 . Lemma 1: Let a b e a p ositi ve integer and G a be a COD of size 2 a × 2 a in ( a + 1 ) complex variables x 1 , · · · , x a +1 as given in (1). Let i be a positive integer between 0 and 2 a − 1 . Let the radix -2 representation of i be ( i a − 1 , i a − 2 , · · · , i 0 ) wher e i a − 1 is the most significant bit. Th en N ( a ) i = { i } ∪ { i + ( − 1) i j 2 j | j = 0 , · · · , a − 1 } or equiv alently , N ( a ) i = { i } ∪ { i ⊕ 2 j | j = 0 to a − 1 } where ⊕ denotes the com ponent- wise m odule 2 additio n of the rad ix-2 representatio n vecto rs. Pr oof: The proo f is by in duction on a . The case a = 1 , correspo nds to the Alam outi co de G 1 . W e n ote that N (1) 0 = { 0 , 1 } and N (1) 1 = { 0 , 1 } as given by th e expression o f N (1) i for i = 0 , 1 . So for a = 1 , the le mma is true. Let the lem ma be true for all a ≤ n . T hen, we hav e N ( n ) i = { i } ∪ { i + ( − 1) i j 2 j | j = 0 , · · · , n − 1 } (7) for all i = 0 , 1 , · · · , 2 n − 1 an d we need to p rove that N ( n +1) i = { i } ∪ { i + ( − 1) i j 2 j | j = 0 , · · · , n } (8) for all i = 0 , 1 , · · · , 2 n +1 − 1 . For a = n + 1 , we have th e radix-2 represen tation, i = ( i n , i n − 1 , · · · , i 0 ) and G n +1 =  G n − x ∗ n +2 I 2 n x n +2 I 2 n G H n  . (9) W e hav e the following two cases: Case (i) 0 ≤ i ≤ 2 n − 1 : In this case i n = 0 and th e term i + ( − 1) i n 2 n in (8) corr esponds to the n on-zero lo cation in the − x ∗ n +2 I 2 n part of G n +1 and th e nonze ro locatio ns in th e G n part is given by the rem aining elements o f (8) which is nothing but N ( n ) i . Case (ii) 2 n ≤ i ≤ 2 n +1 − 1 : In th is case i n = 1 for all values of i in the r ange unde r consideration . Then, the term correspo nding to j = n in (8) is i − 2 n which co rrespond s to the non- zero te rm in the x n +2 I 2 n part of the matrix (9). Also, ev ery term of th e form i + ( − 1 ) i j 2 j ; j = 0 , 1 , · · · , n − 1 in (8) will be same as a term in ( 7) with 2 n added to it. T his takes into acco unt all the no n-zero entries in the G H n part o f (9). Example 2.1: I n this examp le we compute the sets N ( a ) i for a = 2 an d 3 . For a = 2 , the possible values of i are 0 , 1 , 2 4 IEEE TRANSACTIONS ON WIREL ESS COMMUNICA TIONS , VOL. XX, NO. XX , XXXX T ABLE I M a A N D M ′ a F O R a = 3 , · · · 9 a 3 4 5 6 7 8 9 M a { 3 } { 3 } { 3 , 5 } { 3 , 5 , 6 } { 3 , 5 , 6 , 7 } { 3 , 5 , 6 , 7 } { 3 , 5 , 6 , 7 , 9 } M ′ a { 7 } { 7 } { 7 , 25 } { 7 , 25 , 42 } { 7 , 25 , 42 , 75 } { 7 , 25 , 42 , 75 } { 7 , 25 , 42 , 75 , 385 } d 2 3 3 3 3 4 4 and 3 , while for a = 3 , i takes value between 0 and 7 . N (2) 0 = { 0 } ∪ { 0 ⊕ 2 0 , 0 ⊕ 2 1 } = { 0 , 1 , 2 } , N (2) 1 = { 1 } ∪ { 1 ⊕ 2 0 , 1 ⊕ 2 1 } = { 1 , 0 , 3 } , N (2) 2 = { 2 } ∪ { 2 ⊕ 2 0 , 2 ⊕ 2 1 } = { 2 , 3 , 0 } , N (2) 3 = { 3 } ∪ { 3 ⊕ 2 0 , 3 ⊕ 2 1 } = { 3 , 2 , 1 } . N (3) 0 = { 0 } ∪ { 1 , 2 , 4 } = { 0 , 1 , 2 , 4 } , N (3) 1 = { 1 } ∪ { 0 , 3 , 5 } = { 1 , 0 , 3 , 5 } , N (3) 2 = { 2 } ∪ { 3 , 0 , 6 } = { 2 , 3 , 0 , 6 } , N (3) 3 = { 3 } ∪ { 2 , 1 , 7 } = { 3 , 2 , 1 , 7 } , N (3) 4 = { 4 } ∪ { 5 , 6 , 0 } = { 4 , 5 , 6 , 0 } , N (3) 5 = { 5 } ∪ { 4 , 7 , 1 } = { 5 , 4 , 7 , 1 } , N (3) 6 = { 6 } ∪ { 7 , 4 , 2 } = { 6 , 7 , 4 , 2 } , N (3) 7 = { 7 } ∪ { 6 , 5 , 3 } = { 7 , 6 , 5 , 3 } . Notice that N (3) i ∩ N (3) j = φ if i ⊕ j = 7 , where φ repr esents the empty set. Definition 1: T wo rows R i , R j of G a are said to be no n- intersecting if N ( a ) i ∩ N ( a ) j = φ . The follo wing lem ma is needed to prove Lemma 4 which in turn is u sed in the pr oof of the main result given in The orem 1. Lemma 2: Let V a denote the set of all rad ix-2 re presenta- tion vectors of the elements o f the set { 0 , 1 , · · · , 2 a − 1 } and S be a subset of V a . Then N ( a ) i ∩ N ( a ) j = φ fo r all i, j ∈ S, i 6 = j if and on ly if the minim um Ha mming distance (MHD) of S is greater than or equ al to 3 . Pr oof: W e first show that N ( a ) i ∩ N ( a ) j = φ for all i, j ∈ S, i 6 = j implies that the MHD of S is g reater than or equal to 3 . E quiv alently , if the MHD of S is 1 or 2 , th en th ere exists i, j ∈ S, i 6 = j , such that N ( a ) i ∩ N ( a ) j 6 = φ . Assume that the MHD o f S is 1 or 2 , then their exists i, j ∈ S , i 6 = j , such that dist ( i, j ) = 1 o r 2 . So, either i = j ⊕ 2 k for some k ∈ { 0 , 1 , · · · , a − 1 } if dist ( i, j ) = 1 , or i = j ⊕ 2 k 1 ⊕ 2 k 2 for so me k 1 , k 2 ∈ { 0 , 1 , · · · , a − 1 } , k 1 6 = k 2 if dist ( i, j ) = 2 . In the first case, i ∈ N ( a ) i ∩ N ( a ) j and in the seco nd case, ( i ⊕ 2 k 1 ) ∈ N ( a ) i ∩ N ( a ) j as i ⊕ 2 k 1 = j ⊕ 2 k 2 . For bo th ca ses, N ( a ) i ∩ N ( a ) j 6 = φ . Next we prove that if the MHD of S is at least 3 , then N ( a ) i ∩ N ( a ) j = φ for all i, j ∈ S ; i 6 = j , o r equ i valently , if fo r some i, j ∈ S, i 6 = j , N ( a ) i ∩ N ( a ) j 6 = φ , th en MHD o f S is less than 3. Let i, j ∈ S and i 6 = j . W e h a ve N ( a ) i = { i ⊕ 2 k | k = 0 , · · · , a − 1 } ∪ { i } an d N ( a ) j = { j ⊕ 2 k | k = 0 , · · · , a − 1 } ∪ { j } . As N ( a ) i ∩ N ( a ) j 6 = φ , let x ∈ N ( a ) i ∩ N ( a ) j . W e h av e x = i o r x = i ⊕ 2 k 1 for some 0 ≤ k 1 ≤ a − 1 , as x ∈ N ( a ) i . Similarly , x = j o r x = j ⊕ 2 k 2 for some 0 ≤ k 2 ≤ a − 1 , as x ∈ N ( a ) j . But if x = i , then x 6 = j , as i 6 = j . So , we have following three cases: (i) x = i and x = j ⊕ 2 k 2 , (ii) x = i ⊕ 2 k 1 and x = j , (iii) x = i ⊕ 2 k 1 and x = j ⊕ 2 k 2 , k 1 6 = k 2 (as i 6 = j ). For the case (i) & (ii), we have i = j ⊕ 2 k 2 & i ⊕ 2 k 1 = j respectively and in both cases, dist ( i, j ) = 1 . For the case ( iii), we hav e i ⊕ 2 k 1 = j ⊕ 2 k 2 , which means the dist ( i, j ) = 2 . So MHD of S is less than 3. For a given a, let d be the p ositi ve integer such that 2 d − 1 ≤ a < 2 d and a = P d − 1 j =0 a j 2 j , a j ∈ F 2 . Note that a d − 1 = 1 . Define M a = { 0 < x ≤ a | x 6 = 2 k for any k = 0 , 1 , · · · } (10 ) and M ′ a = n 2 x − 1 + d − 1 X j =0 x j 2 2 j − 1 ˛ ˛ ˛ x = d − 1 X j =0 x j 2 j ∈ M a , x j ∈ F 2 o . (11) Note th at the nu mber o f elemen ts in M a is a − d . Moreover , M a ⊆ M b and M ′ a ⊆ M ′ b whenever a ≤ b . When a is a power of 2 , M a = M a − 1 and M ′ a = M ′ a − 1 . Example 2.2: T he sets M a and M ′ a for a = 3 to 9 are shown in T able I at the top o f th is page. Lemma 3: Let M ′ a be as defined in ( 11). V ie w M ′ a as a subset of V = F a 2 by id entifying each elemen t of M ′ a with its radix-2 r epresentation vector of length a . T hen the MHD of the linear space spann ed by M ′ a , deno ted b y S , is 3 . Pr oof: Th e sub space S ⊂ V is given by S = { P a − d − 1 j =0 c j y ′ j | y ′ j ∈ M ′ a } where c j ∈ F 2 for j = 0 , 1 , · · · , a − d − 1 . Observe that th e map g i ven by f : M a → M ′ a x = P d − 1 j =0 x j 2 j 7→ x ′ = 2 x − 1 + P d − 1 j =0 x j 2 2 j − 1 , (12) is one-on e. Thu s, the size of M ′ a , denoted as | M a ′ | is also a − d . Notice th at 2 x − 1 6 = 2 2 j − 1 for j = 0 , 1 , · · · , d − 1 as x 6 = 2 j . So , w t ( x ′ ) = 1 + w t ( x ) fo r all x ∈ M a where w t ( x ) stands for th e Hamming weight of x . Now wt ( x ) ≥ 2 a s x is not a power of 2 . So wt ( x ′ ) ≥ 3 . Similarly , w t ( x ′ ⊕ y ′ ) = 2 + w t ( x ⊕ y ) for all x, y ∈ M a , x 6 = y . Now w t ( x ⊕ y ) ≥ 1 a s x 6 = y , which imp lies that w t ( x ′ ⊕ y ′ ) ≥ 3 fo r a ll x ′ , y ′ ∈ M ′ a . In general, wt ( y ′ 1 ⊕ y ′ 2 ⊕ · · · ⊕ y ′ k ) = k + wt ( y 1 ⊕ y 2 ⊕ · · · ⊕ y k ) for k ≤ a − d , y ′ 1 6 = y ′ 2 6 = · · · 6 = y ′ k . So f or a ll k ≥ 3 and k ≤ a − d , wt ( y ′ 1 ⊕ y ′ 2 ⊕ · · · ⊕ y ′ k ) ≥ 3 . Now there exists DAS and RAJ AN: SQUARE COMPLEX OR THOGONAL DESIGNS WITH LOW P APR AND SIGNALING COMPLEXITY 5 an element in S , for instance 7 , wh ose Hammin g weigh t is 3 . Thus, the MHD of S is 3 . Lemma 4: Let a an d d be n on-zero p ositi ve integers such that 2 d − 1 ≤ a < 2 d and V a = { 0 , 1 , · · · , 2 a − 1 } . Then , there exists a partition of V a into 2 d subsets C ( a ) j , j = 0 , 1 , · · · , 2 d − 1 each containing 2 a − d elements, such that for any two distinct elements x, y ∈ C ( a ) j , j ∈ { 0 , 1 , · · · , 2 d − 1 } , we have N ( a ) x ∩ N ( a ) y = φ. Pr oof: W e ide ntify the set V a with F a 2 by viewing each element of V a with its radix-2 representation vector . Let M ′ a be as given by ( 11) and S be the sub-space of V a spanned by the radix-2 rep resentation vectors (of length a ) o f the elemen ts of the set M ′ a . The nu mber of elements in S is 2 a − d . By Lemma 3, the MHD of S is 3 . Now we d efine a relation ′ ∼ ′ on V a as follows: For all a, b ∈ V a , a ∼ b , if a ⊕ b ∈ S . W e observe that this is an equiv alence r elation as for all a, b and c ∈ V a , 1) a ∼ a as a ⊕ a = 0 ∈ S , 2) a ∼ b ⇒ b ∼ a as a ⊕ b ∈ S, imp lies that b ⊕ a ∈ S . 3) a ∼ b and b ∼ c, then a ∼ c, as a ⊕ b ∈ S and b ⊕ c ∈ S, together imply a ⊕ c ∈ S . The n umber o f eq uiv alence classes is 2 a 2 a − d = 2 d and these equiv alence classes are deno ted a s C ( a ) i , i = 0 , 1 , · · · , 2 d − 1 . For any o ne equiv alence class C ( a ) i , th e elem ents in C ( a ) i are giv en by { x ⊕ s | s ∈ S } f or som e x ∈ C ( a ) i . Now the MHD of the class C ( a ) i , is also equa l to th e MHD of S which is 3. By lemma 2, N ( a ) x ∩ N ( a ) y = φ for all x, y ∈ C ( a ) i , i = 0 to 2 d − 1 . The f ollowing example illu strates th e partitio n of V a into the subsets C ( a ) i , i = 0 to 2 d − 1 , for a = 3 , 4 , 5 and 6 . Example 2.3: (i) L et a = 3 . V 3 is p artitioned in to 4 classes C (3) 0 , C (3) 1 , C (3) 2 and C (3) 3 , each containing 2 elements. W e have already seen that M 3 = { 3 } and M ′ 3 = { 7 } . C ( a ) i = { i, i ⊕ 7 } for i = 0 , 1 , 2 and 3 . Explicitly , C (3) 0 = { 0 , 7 } , C (3) 1 = { 1 , 6 } , C (3) 2 = { 2 , 5 } , C (3) 3 = { 3 , 4 } . (ii) For a = 4 , C (4) 0 = { 0 , 7 } , C (4) 1 = { 1 , 6 } , C (4) 2 = { 2 , 5 } , C (4) 3 = { 3 , 4 } , C (4) 4 = { 8 , 15 } , C (4) 5 = { 9 , 14 } , C (4) 6 = { 10 , 13 } , C (4) 7 = { 11 , 12 } . (iii) For a = 5 , C (5) 0 = { 0 , 7 , 25 , 3 0 } , C (5) 1 = { 1 , 6 , 24 , 3 1 } , C (5) 2 = { 2 , 5 , 27 , 2 8 } , C (5) 3 = { 3 , 4 , 26 , 2 9 } , C (5) 4 = { 8 , 15 , 17 , 22 } , C (5) 5 = { 9 , 1 4 , 16 , 23 } , C (5) 6 = { 10 , 13 , 1 9 , 20 } , C (5) 7 = { 11 , 12 , 1 8 , 21 } . (iv) For a = 6 , C (6) 0 = { 0 , 7 , 25 , 3 0 , 42 , 45 , 51 , 52 } , C (6) 1 = { 1 , 6 , 24 , 3 1 , 43 , 44 , 50 , 53 } , C (6) 2 = { 2 , 5 , 27 , 2 8 , 40 , 47 , 49 , 54 } , C (6) 3 = { 3 , 4 , 26 , 2 9 , 41 , 46 , 48 , 55 } , C (6) 4 = { 8 , 15 , 17 , 22 , 34 , 3 7 , 59 , 60 } , C (6) 5 = { 9 , 14 , 16 , 23 , 35 , 3 6 , 58 , 61 } , C (6) 6 = { 10 , 13 , 1 9 , 20 , 32 , 39 , 57 , 62 } , C (6) 7 = { 11 , 12 , 1 8 , 21 , 33 , 38 , 56 , 63 } . Theorem 1: Let a an d d be n on-zero po siti ve integers and 2 d − 1 ≤ a < 2 d . There exists a SCOD H a of size 2 a × 2 a with entries of the matr ix 2 a − d 2 H a consisting ± x 1 , ± x 2 , · · · , ± x a +1 or their co njugates, such that the co de has rate R = a +1 2 a and the ratio of num ber of zeros to th e total number o f entries of the matr ix is eq ual to 1 − R · 2 ⌊ log 2 1 R ⌋ . Pr oof: The SCOD H a satisfying the r equired rate an d fraction of zer os in the matrix is o btained fro m the COD G a of size 2 a × 2 a (given in (1)) as f ollows: The rate R of the COD G a is a +1 2 a . Using Lemm a 4, 2 a rows of the COD G a can be partitioned into 2 d group s with each group co ntaining 2 a − d rows such that a ny two distinct rows from any of 2 d group s, is non -intersecting. I f we ad d o r subtract all the rows in a gi ven class, th e resulting r ow will not have any entry which is a linear comb ination of two o r m ore variables. La beling the grou ps as C ( a ) 0 , C ( a ) 1 , · · · , C ( a ) 2 d − 1 , we define the 2 a − d × 2 a matrices B i formed by the rows o f G a which are in C ( a ) i for i = 0 to 2 d − 1 . Now f orm th e matrix B ′ =      B 0 B 1 . . . B 2 d − 1      . (13) The matrix B ′ is of size 2 a × 2 a and it is related to G a by B ′ = PG a where P is a permu tation m atrix of size 2 a × 2 a . W e consider a Hadamard matrix H of size 2 a − d × 2 a − d containing 1 and − 1 such that H T H = 2 a − d I 2 a − d . Let e B i = HB i . The required m atrix H a is H a = 2 − a − d 2      e B 0 e B 1 . . . e B 2 d − 1      . (14) For two m atrices A = [ a ij ] an d B , the tensor produc t of A with B , denoted b y A ⊗ B , is the matrix [ a i,j B ] . Let e H = I 2 d ⊗ H . W e can write H a = 2 − a − d 2 e HB ′ = 2 − a − d 2 e HPG a . Now H a is a SCOD if and o nly if 2 − a − d 2 e HP is an unitar y matrix. As P is per mutation ma trix, it is enough to prove that 2 − a − d 2 e H is an unitary matr ix. Indee d, e H T e H = I 2 d ⊗ ( H T H ) = 2 a − d I 2 a , thus H a is a SCOD. The numbe r o f lo cations con taining 0 in any r ow of H a is 2 a − ( a + 1 )2 a − d . Hence the frac tion of zer os in H a is equ al to 2 a − ( a +1)2 a − d 2 a = 1 − a +1 2 a 2 a − d . No w 2 a − d ≤ 2 a a +1 < 2 a − d +1 as 2 d ≥ a + 1 > 2 d − 1 . So, a − d ≤ log 2 2 a a +1 < a − d + 1 and a − d = ⌊ log 2 2 a a +1 ⌋ = ⌊ log 2 1 R ⌋ . Thus the fraction of zero s is 1 − R · 2 ⌊ log 2 1 R ⌋ . The proo f of Theor em 1 suggests a recipe to con struct the SCOD H a with the fra ction of zero s specified in the statement, from a COD G a giv en in (1). The following example illustrates th is r ecipe. Example 2.4: W e consider the co nstruction of rate- 1 / 2 , 8 × 8 COD with no zero in the matr ix. Following the recipe described above, the perm utation matrix P and e H ar e given by 6 IEEE TRANSACTIONS ON WIREL ESS COMMUNICA TIONS , VOL. XX, NO. XX , XXXX P = 2 6 6 6 6 6 6 6 6 4 10 0 0 00 0 0 00 0 0 00 0 1 01 0 0 00 0 0 00 0 0 00 1 0 00 1 0 00 0 0 00 0 0 01 0 0 00 0 1 00 0 0 00 0 0 10 0 0 3 7 7 7 7 7 7 7 7 5 , e H = 2 6 6 6 6 6 6 6 6 4 1 10 00 00 0 1 − 1 0 00 0 0 0 0 01 10 00 0 0 01 − 10 0 0 0 0 00 01 10 0 0 00 01 − 1 0 0 0 00 00 01 1 0 00 01 01 − 1 3 7 7 7 7 7 7 7 7 5 respectively . The matrix H 3 = 2 − 1 2 e HPG 3 is giv en by 1 √ 2 2 6 6 6 6 6 6 6 6 4 x 1 − x ∗ 2 − x ∗ 3 x 4 − x ∗ 4 − x 3 x 2 x ∗ 1 x 1 − x ∗ 2 − x ∗ 3 − x 4 − x ∗ 4 x 3 − x 2 − x ∗ 1 x 2 x ∗ 1 x 4 − x ∗ 3 − x 3 − x ∗ 4 x 1 − x ∗ 2 x 2 x ∗ 1 − x 4 − x ∗ 3 x 3 − x ∗ 4 − x 1 x ∗ 2 x 3 x 4 x ∗ 1 x ∗ 2 − x 2 x 1 − x ∗ 4 x ∗ 3 x 3 − x 4 x ∗ 1 x ∗ 2 x 2 − x 1 − x ∗ 4 − x ∗ 3 x 4 x 3 − x 2 x 1 x ∗ 1 x ∗ 2 x ∗ 3 − x ∗ 4 − x 4 x 3 − x 2 x 1 − x ∗ 1 − x ∗ 2 − x ∗ 3 − x ∗ 4 3 7 7 7 7 7 7 7 7 5 which is a row perm uted version of the cod e g i ven in ( 6). I I I . P R E M U LT I P LYI N G M AT R I X In this section, we presen t a proced ure to comp ute a matrix denoted as Q ( a ) of size 2 a × 2 a which when pre-multiplies G a , along with an ap propr iate scaling factor, th e resulting matr ix H a is the SCOD with d esired fraction of zer os. The scaling factor , when multip lied to the matrix Q ( a ) , m akes it a un itary matrix. In or der to construct th e matrix Q ( a ) , we first associate a 2 x × 2 x matrix Q x to each x in M a . Th e ( i, j ) th e lement of Q x , d enoted by Q x ( i, j ) , is defin ed a s f ollows: For i = 0 to 2 x − 1 − 1 , 1) Q x ( i, i ) = 1; 2) Q x ( i, i ⊕ x ′ ) = 1 ; 3) Q x ( i ⊕ x ′ , i ) = 1; 4) Q x ( i ⊕ x ′ , i ⊕ x ′ ) = − 1; 5) Q x ( i, j ) = 0 for all oth er values of i an d j ; where x ′ is given by (12). Define a 2 a × 2 a matrix e Q x as e Q x = I 2 a − x ⊗ Q x where I 2 a − x is th e identity m atrix o f size 2 a − x × 2 a − x . Example 3.1: In this e xample, we comp ute Q x for x ∈ M 4 . W e have M 4 = { 3 } . The matr ix Q 3 is the matrix sh own on the left hand side of ( 6) without th e scaling factor 1 √ 2 . T ow ards the co nstruction o f p remultiplyin g matrix, we need the following result which say s that e Q x and e Q y commute fo r all x, y ∈ M a . Lemma 5: e Q x e Q y = e Q y e Q x for all x, y ∈ M a . Pr oof: Let the i th row of e Q x be e Q i x . There are 2 a rows and 2 a columns f or th e matr ix e Q x with n on-zero entr ies either 1 or − 1 . Fix i . Let e Q x ( i, l ) = c l , c l ∈ { 0 , 1 , − 1 } for l = 0 to 2 a − 1 . So e Q i x = ( c 0 , c 1 , · · · , c 2 a − 1 ) . W e wr ite ( c 0 , c 1 , · · · , c 2 a − 1 ) as P 2 a − 1 l =0 c l 2 l and this corre spondenc e is un ique in the sense that no two d istinct vector s will prod uce the same value unde r the above co rrespond ence. Then, we have e Q i x = P 2 a − 1 l =0 c l 2 l . Let the radix- 2 representatio n of i and j be ( i a − 1 , i a − 2 , · · · , i 0 ) and ( j a − 1 , j a − 2 , · · · , j 0 ) respectively . Note th at e Q i x = ( − 1) i x − 1 2 i + 2 i ⊕ x ′ for i = 0 to 2 a − 1 . Moreover , e Q x is a symmetric matrix for all x ∈ M a . Let K = e Q x e Q y and the ( i, j ) th entry of K b e K ( i, j ) . It follows that K ( i, j ) =                0 if i ⊕ j / ∈ { 0 , x ′ , y ′ , x ′ ⊕ y ′ } ( − 1) i x − 1 if i ⊕ j = y ′ ( − 1) j y − 1 if i ⊕ j = x ′ 1 if i ⊕ j = x ′ ⊕ y ′ ( − 1) i x − 1 + j y − 1 if i ⊕ j = 0 , K ( j, i ) =                0 if i ⊕ j / ∈ { 0 , x ′ , y ′ , x ′ ⊕ y ′ } ( − 1) j x − 1 if i ⊕ j = y ′ ( − 1) i y − 1 if i ⊕ j = x ′ 1 if i ⊕ j = x ′ ⊕ y ′ ( − 1) j x − 1 + i y − 1 if i ⊕ j = 0 . Let x = P d − 1 l =0 x l 2 l and y = P d − 1 l =0 y l 2 l . W e h av e y ′ = 2 y − 1 + P d − 1 l =0 y l 2 2 l − 1 . Note that th e ( x − 1 ) th compo nent o f y ′ , i.e., the coefficient of 2 x − 1 in the radix-2 rep resentation of y ′ is zero as x 6 = y and x is n ot a power of 2 . Thus i x − 1 = j x − 1 when i ⊕ j = y ′ . Similarly j y − 1 = i y − 1 when i ⊕ j = x ′ and i x − 1 + j y − 1 = j x − 1 + i y − 1 when i ⊕ j = 0 , i.e., i = j . Thus K ( i, j ) = K ( j, i ) for all i, j ∈ { 0 , 1 , · · · , 2 a − 1 } . So, K = e Q x e Q y is sym metric. Th en, K T = e Q T y e Q T x = K = e Q x e Q y . As e Q y and e Q x are symm etric m atrix, they commu te. Let Q ( a ) = Q x ∈ M a e Q x , which is well defin ed sin ce the produ ct o f th ese matrices does not depend on the order these matrices are multiplied . The fo llowing theorem asserts that Q ( a ) so constructed , will pro duce a SCOD with th e de sired fraction of zeros. Theorem 2: Let a an d d b e non- zero positiv e integers and 2 d − 1 ≤ a < 2 d . Then H a = 2 − a − d 2 Q ( a ) G a is th e d esired SCOD with th e rate and th e fraction of zer os as spe cified in Theorem 1. Pr oof: W e have to pr ove that H a = 2 − a − d 2 Q ( a ) G a is a SCOD o f size 2 a × 2 a with rate R = a +1 2 a and the fractio n of ze ros is equal to 1 − R · 2 ⌊ log 2 1 R ⌋ . Since we obtain H a by pre -multiplyin g a constant matrix to G a , the rate remains same. T hus, it is enough to prove tha t H a is a SCOD and it contains the desired fraction of zeros. As G a is a COD, so H a is a COD if 2 − a − d 2 Q ( a ) is an unitary matrix . Moreover , if each row o f H a contains 2 a − d ( a + 1) n on-zero entries, then it contains the required fraction of zeros. It is easy to note that if the matr ix Q ( a ) has 2 a − d non-ze ro elemen ts in each o f its rows, then ea ch row of H a will have 2 a − d ( a + 1) n on-zero entries. The co lumn co-ordinates o f the non -zero en tries of i th row of Q ( a ) be such th at the r esulting matr ix H a will not ha ve any entry which is linear co mbination of complex variables, i.e., tho se rows w ill be added or subtr acted which p ossess non-in tersecting property . Thus we have to prove f ollowing two c laims: 1) 2 − a − d 2 Q ( a ) is an unitary matrix; 2) The column co-o rdinates o f no n-zero entries on the i - th row of Q ( a ) is given by the set S a i = { s ⊕ i | s ∈ S a } wher e the subspace S a ⊂ F a 2 is spann ed by the set M ′ a ⊂ F a 2 . DAS and RAJ AN: SQUARE COMPLEX OR THOGONAL DESIGNS WITH LOW P APR AND SIGNALING COMPLEXITY 7 W e know that | M a ′ | is a − d , th us | S a | is 2 a − d . First we prove claim 1). L et x ∈ M a . Con sider the i - th and j -th column of Q x which are denote d as Q i x and Q j x respectively . The inn er p roduct o f Q i x and Q j x , deno ted as h Q i x , Q j x i is as follows: 1) for j 6 = i, i ⊕ x ′ , h Q i x , Q j x i = 0 , 2) for j = i ⊕ x ′ , h Q i x , Q j x i = 1 − 1 = 0 , 3) For j = i , h Q i x , Q j x i = 1 + 1 = 2 . So Q T x Q x = 2 I 2 x where I 2 x is an 2 x × 2 x identity matrix. Now e Q T x e Q x = I 2 a − x ⊗ Q T x Q x which implies that e Q T x e Q x = 2 I 2 a and Q ( a ) T Q ( a ) = 2 a − d I 2 a . Thus, 2 − a − d 2 Q ( a ) is an unitary matrix. Next, we p rove claim 2) by inductio n on a . Let a = 3 , then Q (3) = e Q 3 = Q 3 . Th e i -th row of Q 3 contains no n-zero entries only at i and i ⊕ 7 , correspo nding to the elements of { s ⊕ i | s ∈ S 3 } wh ere S 3 = { 0 , 7 } . Ob serve that S 3 is th e subspace spanned by M ′ 3 = { 7 } ⊂ F 3 2 . Let it be tr ue for a ≤ ( n − 1) . Und er this a ssumption, the co l- umn co-o rdinate of the non- zero en tries on the i th row of the matrix Q ( n − 1) is giv en by th e set S n − 1 i = { s ⊕ i | s ∈ S n − 1 } for i = 0 to 2 n − 1 − 1 . W e have the following two cases: Case ( i) n is a power of 2 : In this case M ′ n = M ′ n − 1 and Q ( n ) = ( I 2 ⊗ Q ( n − 1) ) . fact 2) is tr i vially satisfied. Case (ii) n is not a power of 2 : M n = M n − 1 ∪ { n } and Q ( n ) = e Q n ( I 2 ⊗ Q ( n − 1) ) = Q n ( I 2 ⊗ Q ( n − 1) ) . The i -th row of Q n is given by e Q i n = ( − 1) i n − 1 2 i + 2 i ⊕ n ′ for i = 0 to 2 n − 1 an d j -th row o f Q ( n − 1) , denoted by Q ( n − 1) ,j is P s ∈ S n − 1 j ( − 1) α s 2 s where α s is either +1 or − 1 , dependin g on s . The ( i, j ) -th en try of Q ( n ) is given by the inner prod uct of th e i -th row of Q n with j th row of ( I 2 ⊗ Q ( n − 1) ) as Q ( n − 1) and ( I 2 ⊗ Q ( n − 1) ) both are symmetric matrix. V iewing S n − 1 i ⊂ F n − 1 2 , as a subset of F n 2 , by identify ing F n − 1 2 as a subspace of F n 2 , it is possible to express S n i in term of S n − 1 i . The column co-ordina te of the non-zer o entries at the i -th ro w of Q ( n ) is S n i = S n − 1 i ∪ { s ⊕ n ′ | s ∈ S n − 1 i } fo r i = 0 to 2 n − 1 . Thus S n i = { s ⊕ i | s ∈ S n } . The following theor em shows that the matrix H a with re- quired rate an d the fr action of zeros as specified in Theorem 1, can also be constru cted recursively , in a similar fashion as for G a , and it is do ne using th e premultip lying m atrix Q ( a ) . Theorem 3: Let a be a non -zero po siti ve i nteger and H a be the SCOD as stated in Th eorem 2 . Then H a +1 is constructed recursively u sing H a as follows: H a +1 = ( H ′ a +1 when ( a + 1) is a power o f 2 1 √ 2 Q a +1 H ′ a +1 otherwise , where H ′ a +1 =  H a ( x 1 , x 2 , · · · , x a +1 ) − x ∗ a +2 H a (1 , 0 , · · · , 0) x a +2 H a (1 , 0 , · · · , 0) H a ( x ∗ 1 , − x 2 , · · · , − x a +1 )  . Pr oof: W e hav e M a +1 = ( M a when ( a + 1) is a p ower o f 2 , M a ∪ { a + 1 } otherwise . Now Q ( a ) = Q x ∈ M a e Q x where e Q x is 2 a × 2 a matrix. T o indicate the d ependen ce o f the size of e Q x on a , we write e Q x as e Q a x . W e h av e e Q a x = I 2 a − x ⊗ Q x and e Q a +1 x = I 2 a +1 − x ⊗ Q x . So, e Q a +1 x = I 2 ⊗ e Q a x . Moreover , Q ( a ) = Q x ∈ M a e Q a x and Q ( a +1) = Q x ∈ M a +1 e Q a +1 x . Observe that, if a + 1 is n ot a power o f 2 , then Q ( a +1) = e Q a +1 a +1 · ( I 2 ⊗ Q ( a ) ) . Since e Q a +1 a +1 = Q a +1 , we have Q ( a +1) = ( I 2 ⊗ Q ( a ) when ( a + 1) is a p ower of 2 , Q a +1 · ( I 2 ⊗ Q ( a ) ) otherwise . From Theor em 2 , we have H a = 2 − a − d 2 Q ( a ) G a where d is giv en b y 2 d − 1 ≤ a < 2 d . Hence, H a +1 = ( 2 − a − d 2 · ( I 2 ⊗ Q ( a ) ) · G a +1 when ( a + 1) i s a power of 2 , 2 − a − d +1 2 · Q a +1 · ( I 2 ⊗ Q ( a ) ) · G a +1 otherwise. Let H ′ a +1 = 2 − a − d 2 ( I 2 ⊗ Q ( a ) ) G a +1 . (15) Then, H a +1 = ( H ′ a +1 when ( a + 1) is a power o f 2 , 1 √ 2 Q a +1 H ′ a +1 otherwise . H ′ a +1 is constructed using H a as follows: W e hav e from (1), G a +1 =  G a − x ∗ a +2 I 2 a x a +2 I 2 a G H a  . (16) From the constructio n of G a and H a , it follows that I 2 a = G a (1 , 0 , · · · , 0) , Q ( a ) = 2 a − d 2 H a (1 , 0 , · · · , 0) , G H a = G H a ( x 1 , x 2 , · · · , x a +1 ) = G a ( x ∗ 1 , − x 2 , · · · , − x a +1 ) . (1 7) Using (15), (16) and (1 7), we h av e H ′ a +1 = H ′ a +1 ( x 1 , x 2 , · · · , x a +2 ) = » H a ( x 1 , x 2 , · · · , x a +1 ) − x ∗ a +2 H a (1 , 0 , · · · , 0) x a +2 H a (1 , 0 , · · · , 0) H a ( x ∗ 1 , − x 2 , · · · , − x a +1 ) – . For a = 3 , Q (3) is given in (6). For a = 4 , Q (4) is as follows: 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 1 0 00 0 0 0 1 0 0 0 0 0 0 0 0 0 1 00 0 0 1 0 0 0 0 0 0 0 0 0 0 0 10 0 1 0 0 0 0 0 0 0 0 0 0 0 0 01 1 0 0 0 0 0 0 0 0 0 0 0 0 0 01 − 0 0 0 0 0 00 0 0 0 0 0 0 10 0 − 0 0 0 0 0 0 0 0 0 0 0 1 00 0 0 − 0 0 0 0 0 0 0 0 0 1 0 00 0 0 0 − 00 00 0 0 0 0 0 0 00 0 0 0 0 1 0 0 0 0 0 0 1 0 0 00 0 0 0 0 0 1 0 0 0 0 1 0 0 0 00 0 0 0 0 0 0 1 0 0 1 0 0 0 0 00 0 0 0 0 0 0 0 1 1 0 0 0 0 0 00 0 0 0 0 0 0 0 1 − 0 0 0 0 0 00 0 0 0 0 0 0 1 0 0 − 0 0 0 0 00 0 0 0 0 0 1 0 0 0 0 − 0 0 0 00 0 0 0 0 1 0 0 0 0 0 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 5 The SCOD obtained b y premultiply ing G 4 with 2 − 1 2 Q (4) , is shown on the lef t h and side at th e to p of the next page. 8 IEEE TRANSACTIONS ON WIREL ESS COMMUNICA TIONS , VOL. XX, NO. XX , XXXX T ABLE II C O M PA R I S O N O F P OW E R D I S T R I B U T I O N C H A R AC T E R I S T I C S 16 Tx; QPS K 16 Tx; 16 QAM 32 Tx; QPS K 32 T x; 16 QAM SCODs G a Codes in thi s paper Peak/a ve P 0 3.2 0.6875 1.6 0.375 Peak/a ve P 0 11 . 52 0 . 6875 5 . 76 0 . 375 Peak/a ve P 0 5 . 33 0 . 8125 1 . 33 0 . 25 Peak/a ve P 0 19 . 2 0 . 8125 4 . 8 0 . 25 T ABLE III V A R I A T I O N O F F R A C T I O N O F ZE RO S W I T H T H E N U M B E R O F A N T E N N A S a 3 4 5 6 7 8 9 10 11 12 13 14 15 16 f z ( H a ) 0 3 8 2 8 1 8 0 7 16 6 16 5 16 4 16 3 16 2 16 1 16 0 15 32 f z ( G a ) 1 / 2 11 16 13 16 57 64 120 128 247 256 502 512 1013 1024 2036 2048 4083 4096 8178 8192 16369 16384 32752 32768 65519 65536 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 4 x 1 − x ∗ 2 − x ∗ 3 x 4 − x ∗ 4 − x 3 x 2 x ∗ 1 − x ∗ 5 0 0 0 0 0 0 − x ∗ 5 x 2 x ∗ 1 x 4 − x ∗ 3 − x 3 − x ∗ 4 x 1 − x ∗ 2 0 − x ∗ 5 0 0 0 0 − x ∗ 5 0 x 3 x 4 x ∗ 1 x ∗ 2 − x 2 x 1 − x ∗ 4 x ∗ 3 0 0 − x ∗ 5 0 0 − x ∗ 5 0 0 x 4 x 3 − x 2 x 1 x ∗ 1 x ∗ 2 x ∗ 3 − x ∗ 4 0 0 0 − x ∗ 5 − x ∗ 5 0 0 0 − x 4 x 3 − x 2 x 1 − x ∗ 1 − x ∗ 2 − x ∗ 3 − x ∗ 4 0 0 0 − x ∗ 5 x ∗ 5 0 0 0 x 3 − x 4 x ∗ 1 x ∗ 2 x 2 − x 1 − x ∗ 4 − x ∗ 3 0 0 − x ∗ 5 0 0 x ∗ 5 0 0 x 2 x ∗ 1 − x 4 − x ∗ 3 x 3 − x ∗ 4 − x 1 x ∗ 2 0 − x ∗ 5 0 0 0 0 x ∗ 5 0 x 1 − x ∗ 2 − x ∗ 3 − x 4 − x ∗ 4 x 3 − x 2 − x ∗ 1 − x ∗ 5 0 0 0 0 0 0 x ∗ 5 x 5 0 0 0 0 0 0 x 5 x ∗ 1 x ∗ 2 x ∗ 3 − x 4 x ∗ 4 x 3 − x 2 x 1 0 x 5 0 0 0 0 x 5 0 − x 2 x 1 − x 4 x ∗ 3 x 3 x ∗ 4 x ∗ 1 x ∗ 2 0 0 x 5 0 0 x 5 0 0 − x 3 − x 4 x 1 − x ∗ 2 x 2 x ∗ 1 x ∗ 4 − x ∗ 3 0 0 0 x 5 x 5 0 0 0 − x 4 − x 3 x 2 x ∗ 1 x 1 − x ∗ 2 − x ∗ 3 x ∗ 4 0 0 0 x 5 − x 5 0 0 0 x 4 − x 3 x 2 x ∗ 1 − x 1 x ∗ 2 x ∗ 3 x ∗ 4 0 0 x 5 0 0 − x 5 0 0 − x 3 x 4 x 1 − x ∗ 2 − x 2 − x ∗ 1 x ∗ 4 x ∗ 3 0 x 5 0 0 0 0 − x 5 0 − x 2 x 1 x 4 x ∗ 3 − x 3 x ∗ 4 − x ∗ 1 − x ∗ 2 x 5 0 0 0 0 0 0 − x 5 x ∗ 1 x ∗ 2 x ∗ 3 x 4 x ∗ 4 − x 3 x 2 − x 1 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 5 , 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 4 x 1 − x ∗ 2 − x ∗ 3 0 − x ∗ 4 0 0 0 − x ∗ 5 0 0 0 0 0 0 0 x 2 x ∗ 1 0 − x ∗ 3 0 − x ∗ 4 0 0 0 − x ∗ 5 0 0 0 0 0 0 x 3 0 x ∗ 1 x ∗ 2 0 0 − x ∗ 4 0 0 0 − x ∗ 5 0 0 0 0 0 0 x 3 − x 2 x 1 0 0 0 − x ∗ 4 0 0 0 − x ∗ 5 0 0 0 0 x 4 0 0 0 x ∗ 1 x ∗ 2 x ∗ 3 0 0 0 0 0 − x ∗ 5 0 0 0 0 x 4 0 0 − x 2 x 1 0 x ∗ 3 0 0 0 0 0 − x ∗ 5 0 0 0 0 x 4 0 − x 3 0 x 1 − x ∗ 2 0 0 0 0 0 0 − x ∗ 5 0 0 0 0 x 4 0 − x 3 x 2 x ∗ 1 0 0 0 0 0 0 0 − x ∗ 5 x 5 0 0 0 0 0 0 0 x ∗ 1 x ∗ 2 x ∗ 3 0 x ∗ 4 0 0 0 0 x 5 0 0 0 0 0 0 − x 2 x 1 0 x ∗ 3 0 x ∗ 4 0 0 0 0 x 5 0 0 0 0 0 − x 3 0 x 1 − x ∗ 2 0 0 x ∗ 4 0 0 0 0 x 5 0 0 0 0 0 − x 3 x 2 x ∗ 1 0 0 0 x ∗ 4 0 0 0 0 x 5 0 0 0 − x 4 0 0 0 x 1 − x ∗ 2 − x ∗ 3 0 0 0 0 0 0 x 5 0 0 0 − x 4 0 0 x 2 x ∗ 1 0 − x ∗ 3 0 0 0 0 0 0 x 5 0 0 0 − x 4 0 x 3 0 x ∗ 1 x ∗ 2 0 0 0 0 0 0 0 x 5 0 0 0 − x 4 0 x 3 − x 2 x 1 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 5 For co mparison , on the righ t hand side, we h a ve displayed the SCOD G 4 to comp are the n umber of zeros. Th e pr emultiply- ing m atrix Q (5) for 32 ante nnas corresp onding to a = 5 is displayed in Fig. 5 and the resulting co de H 5 in Fig. 6. I V . P A P R O F T H E N E W C O D E S In T able II, the SCODs con structed in this p aper are compare d to the known SCODs given in ( 1). For some fixed number of tran smit a ntennas, we observe that the P APR of these n ew code s is less compare d to the known SCODs. In fact, it is easily seen th at as the num ber of transmit antenna increases, these co des o utperfo rm the existing CODs significantly as far as P APR is concer ned. Quantitatively , for QAM sign al set, the P APR of a SCOD for 2 a antennas giv en by (1), is 2 a a +1 , wh ile it is 2 a ( a +1) · 2 ⌊ log 2 ( 2 a a +1 ) ⌋ for th e SCOD of same size constructed in th is paper . The codes construc ted in this pa per con tain fewer zero s than the well-known SCODs. Hence, the pro bability P 0 that an anten na transmits a zer o symbol ( or switched off) is less in these co des comp ared to the codes given in (1). Another interesting fact to n ote is th at the new SCODs for 2 a transmit anten nas, con tains no zero entry when a + 1 is a power of 2 . F or all o ther v alues o f a , th e fr action of zero s k eeps reducing starting from a = 2 l to 2 l +1 − 1 , l a positive integer . On th e o ther han d, the fraction o f zeros keeps incr easing as a ( ≥ 3 ) increases for the codes gi ven in (1). T able III sh ows the variation in f raction o f zero s (d enoted as f z ) f or pr oposed codes H a and the SCODs G a for a = 3 to 16 . V . S I M U L AT I O N R E S U LT S The symb ol error p erform ance of th e SCODs constructed in this p aper (deno ted as RZCOD in the plots which means COD with Reduced nu mber of Z eros) f or 8 , 16 , 3 2 and 64 antennas are comp ared with that of well-k nown COD (denoted as SCOD) of same o rder in Fig. 1 an d Fig. 3 un der peak power constraint. Similarly , Fig. 2 and Fig. 4 compar e the corre- sponding co des un der average power con straint. The average power con straint perfo rmance of RZCOD matches with that of the comp arable SCOD, while the RZCOD perform s better than the c orrespon ding SCOD under peak p ower constraint as seen in the figur es. W e a lso observe that the per forman ce of our co de fo r 8 transmit a ntennas matches with that o f the code G Y (denoted as Y uen (8)) con structed b y Y uen et al. V I . D I S C U S S I O N W e have given con struction for rate a +1 2 a SCODs for 2 a antennas, for all values of a , with lesser nu mber of zero entries than the known constructio ns. When a + 1 is a power of 2, our c onstruction g i ves SCODs with n o z ero entr ies. Th is case alone g eneralizes th e constru ctions in [7], [8], [9], [1 0] whic h are only fo r 8 an tennas. Some of the possible directio ns for further research are listed below: • For arbitrary v alues of a the fraction of zero entries in ou r codes is 1 − a +1 2 a 2 ⌊ log 2 ( 2 a a +1 ) ⌋ . W e con jecture that SCODs with smaller f raction of zero entries d o not e xist. It will be an interesting d irection to p ursue to settle this conjecture. • Sev eral de signs inc luding CODs h av e been fo und useful in systems explo iting coope rativ e di versity . I t will b e interesting to in vestigate th e su itability of the codes of this paper for coop erativ e diversity . • W e have e xploited the com binatorial structure of the ro ws of the design in ( 1) to obtain the codes with low P APR. The interrelatio nship between A ODs and o ur codes is an importan t direction to pursue. DAS and RAJ AN: SQUARE COMPLEX OR THOGONAL DESIGNS WITH LOW P APR AND SIGNALING COMPLEXITY 9 Fig. 1. The performance of the RZCODs and SCODs for 8, 16 and 32 transmit antennas and the code giv en in Y uen et al for 8 transmit antenna s using QAM modulati on. 0 2 4 6 8 10 12 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 AvgSNR(dB) Symbol error Probability RZCOD(8) SCOD(8Tx) SCOD(16) RZCOD(16) SCOD(32) RZCOD(32) Fig. 2. The performance of the RZCODs and SCODs for 8, 16 and 32 transmit a ntennas using QAM modu lation. 0 1 2 3 4 5 6 10 −8 10 −7 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 PeakSNR(dB) Symbol error Probability SCOD(64) RZCOD(64) Fig. 3. The perfo rmance of the RZCODs and SCODs for 64 transmit antenna s using QAM m odulat ion. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 10 −7 10 −6 10 −5 10 −4 10 −3 10 −2 AvgSNR(dB) Symbol error Probability SCOD(64) RZCOD(64) Fig. 4. The perfo rmance of the RZCODs and SCODs for 64 transmit antenna s using QAM m odulat ion. R E F E R E N C E S [1] V . T arokh, H. Jaf arkhani, and A. R. Cal derbank, “Space -time block codes from orth ogonal designs, ” IEEE T rans. Inf orm. Theory , vol. 45, pp. 1456- 1467, July 19 99. [2] O. Ti rkkonen and A. Hottinen, “Square matrix embeddable STBC for comple x signal constell ations Space-time block codes from orthogonal design, ” IEEE T rans. Inf orm. Theory , V ol 48,no. 2, pp . 384-395, Feb . 2002. [3] X. B . Liang “Ortho gonal Designs with Maximal Rates, ” IEE E T ra ns.Inform. Theory , V ol.49, pp. no. 10, 2468-2503, Oct. 2003. [4] J. F . Adams, P . D. L ax, and R. S. Phillips, “On m atric es whose real linear combinat ions are nonsingular , ” Pr oc. A mer . Math. Soc., vol. 16, 1965, pp. 318-3 22. [5] T . Jozefiak, “ Realiz ation of Hurwitz-Rad on m atric es, ” Queen’ s P apers on Pur e and applied Mathemati cs, vol. 36, pp. 346-351. [6] W . W olfe, Amicabl e Orthogonal Designs-e xistence , Canadian J . Math e- matics, v ol.28, no.5, pp.1006 -1020, 1976. [7] C. Y uen, Y .L. Guan and T . T . Tjhung, “Orthogo nal space time block code from a micable orthogonal design, ” Pr oc. IEEE. Int. Conf . Acoustic, Speec h and Signal , 2004. [8] L. C. Tra n, T . A. W ysocki, A. Mertins and J. Seberry , Complex Ortho g- onal Space-T ime Pr ocessing in wir eless communic ations, Spring er , 2 006. [9] J. Seberry , L. C. Tran, Y . W ang, B. J. W ysocki, T . A. W ysocki, T . Xia and Y . Zhao, “Ne w comple x orthogona l space-time block codes of order eight, ” in T .A.W y socki, B.Honary and B. J .W ycocki., edito rs, Signal Pro cessing for T elecommuni cations and Multimedia , V ol.27 of Multimedi a systems and applicat ions, pp.173-182, Springer , New Y ork, 2004. [10] Y . Z hao, J. Sebe rry , T . Xia, Y . W ang, B. J. W ysoc ki, T . A. W ysocki, L. C. Tran, “ Amic able ort hogonal designs of order 8 for comple x space-ti me block code s, ” T o appe ar in Austr alian J ournal of Comb inatorics (AJC) . [11] A. V . Geramita and J . Seberry , Orthogonal Designs: Quadrati c forms and Hadamar d matrices, V ol.43, Lecture Notes in P ure and Applied Mathemat ics, Marcel Dekker , New Y ork and Ba sel, 1979. [12] Z afa r Ali Khan and B. Sundar Rajan, “Single-Symbol Maximum- Likel ihood Deco dable Line ar STBCs, ” IE EE T rans. Inform. Theory , V ol.52, No.5, Ma y 2006, pp.2062-20 91. [13] C. Y uen, Y . L. Guan and T . T . Tjhung, “Power -Balanc ed Orthogonal Space-T ime Block Code, ” to appe ar in IEEE T rans. V ehicular T echnol ogy . 10 IEEE TRANSACTIONS ON WIREL ESS COMMUNICA TIONS , VOL. XX, NO. XX, XXXX 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 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 − 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 − 0 0 0 0 1 0 0 − 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 − 0 0 0 0 1 0 0 0 0 − 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 − 1 0 0 0 0 0 0 − 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 − 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 − 0 0 0 0 0 1 0 0 − 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 − 0 0 0 0 0 1 − 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 − 0 1 0 0 0 0 0 0 − 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 − 0 1 0 0 0 0 − 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 − 0 0 0 0 0 0 − 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 − 0 0 0 0 − 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 − 0 0 − 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 0 − − 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 − 0 0 0 0 0 − 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 − 0 0 0 0 0 − 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 − 0 − 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 − 0 − 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 − 0 0 0 0 0 0 − 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 − 0 0 0 0 − 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 − 0 0 − 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 − − 0 0 0 0 0 1 0 0 − 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 − 1 0 0 0 0 0 0 1 − 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 − 0 0 1 0 0 1 0 0 0 0 0 0 − 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 − 0 0 0 0 1 0 0 1 0 0 0 0 − 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 − 0 0 0 0 0 0 1 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 Fig. 5. The pre multiplyi ng matrix Q (5) for 32 an tennas 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 4 x 1 − x ∗ 2 − x ∗ 3 x 4 − x ∗ 4 − x 3 x 2 x ∗ 1 − x ∗ 5 x 6 0 0 0 0 x 6 − x ∗ 5 − x ∗ 6 − x 5 0 0 0 0 − x 5 − x ∗ 6 x 2 x ∗ 1 x 4 − x ∗ 3 − x 3 − x ∗ 4 x 1 − x ∗ 2 x 2 x ∗ 1 x 4 − x ∗ 3 − x 3 − x ∗ 4 x 1 − x ∗ 2 x 6 − x ∗ 5 0 0 0 0 − x ∗ 5 x 6 − x 5 − x ∗ 6 0 0 0 0 − x ∗ 6 − x 5 x 1 − x ∗ 2 − x ∗ 3 x 4 − x ∗ 4 − x 3 x 2 x ∗ 1 x 3 x 4 x ∗ 1 x ∗ 2 − x 2 x 1 − x ∗ 4 x ∗ 3 0 0 − x ∗ 5 x 6 x 6 − x ∗ 5 0 0 0 0 − x ∗ 6 − x 5 − x 5 − x ∗ 6 0 0 x 4 x 3 − x 2 x 1 x ∗ 1 x ∗ 2 x ∗ 3 − x ∗ 4 x 4 x 3 − x 2 x 1 x ∗ 1 x ∗ 2 x ∗ 3 − x ∗ 4 0 0 x 6 − x ∗ 5 − x ∗ 5 x 6 0 0 0 0 − x 5 − x ∗ 6 − x ∗ 6 − x 5 0 0 x 3 x 4 x ∗ 1 x ∗ 2 − x 2 x 1 − x ∗ 4 x ∗ 3 − x 4 x 3 − x 2 x 1 − x ∗ 1 − x ∗ 2 − x ∗ 3 − x ∗ 4 0 0 x 6 − x ∗ 5 x ∗ 5 − x 6 0 0 0 0 − x 5 − x ∗ 6 x ∗ 6 x 5 0 0 x 3 − x 4 x ∗ 1 x ∗ 2 x 2 − x 1 − x ∗ 4 − x ∗ 3 x 3 − x 4 x ∗ 1 x ∗ 2 x 2 − x 1 − x ∗ 4 − x ∗ 3 0 0 − x ∗ 5 x 6 − x 6 x ∗ 5 0 0 0 0 − x ∗ 6 − x 5 x 5 x ∗ 6 0 0 − x 4 x 3 − x 2 x 1 − x ∗ 1 − x ∗ 2 − x ∗ 3 − x ∗ 4 x 2 x ∗ 1 − x 4 − x ∗ 3 x 3 − x ∗ 4 − x 1 x ∗ 2 x 6 − x ∗ 5 0 0 0 0 x ∗ 5 − x 6 − x 5 − x ∗ 6 0 0 0 0 x ∗ 6 x 5 x 1 − x ∗ 2 − x ∗ 3 − x 4 − x ∗ 4 x 3 − x 2 − x ∗ 1 x 1 − x ∗ 2 − x ∗ 3 − x 4 − x ∗ 4 x 3 − x 2 − x ∗ 1 − x ∗ 5 x 6 0 0 0 0 x 6 x ∗ 5 − x ∗ 6 − x 5 0 0 0 0 x 5 x ∗ 6 x 2 x ∗ 1 − x 4 − x ∗ 3 x 3 − x ∗ 4 − x 1 x ∗ 2 x 5 x 6 0 0 0 0 x 6 x 5 x ∗ 1 x ∗ 2 x ∗ 3 − x 4 x ∗ 4 x 3 − x 2 x 1 − x 2 x 1 − x 4 x ∗ 3 x 3 x ∗ 4 x ∗ 1 x ∗ 2 − x ∗ 6 x ∗ 5 0 0 0 0 x ∗ 5 − x ∗ 6 x 6 x 5 0 0 0 0 x 5 x 6 − x 2 x 1 − x 4 x ∗ 3 x 3 x ∗ 4 x ∗ 1 x ∗ 2 x ∗ 1 x ∗ 2 x ∗ 3 − x 4 x ∗ 4 x 3 − x 2 x 1 x ∗ 5 − x ∗ 6 0 0 0 0 − x ∗ 6 x ∗ 5 0 0 x 5 x 6 x 6 x 5 0 0 − x 3 − x 4 x 1 − x ∗ 2 x 2 x ∗ 1 x ∗ 4 − x ∗ 3 − x 4 − x 3 x 2 x ∗ 1 x 1 − x ∗ 2 − x ∗ 3 x ∗ 4 0 0 − x ∗ 6 x ∗ 5 x ∗ 5 − x ∗ 6 0 0 0 0 x 6 x 5 x 5 x 6 0 0 − x 4 − x 3 x 2 x ∗ 1 x 1 − x ∗ 2 − x ∗ 3 x ∗ 4 − x 3 − x 4 x 1 − x ∗ 2 x 2 x ∗ 1 x ∗ 4 − x ∗ 3 0 0 x ∗ 5 − x ∗ 6 − x ∗ 6 x ∗ 5 0 0 0 0 x 6 x 5 − x 5 − x 6 0 0 x 4 − x 3 x 2 x ∗ 1 − x 1 x ∗ 2 x ∗ 3 x ∗ 4 − x 3 x 4 x 1 − x ∗ 2 − x 2 − x ∗ 1 x ∗ 4 x ∗ 3 0 0 x ∗ 5 − x ∗ 6 x ∗ 6 − x ∗ 5 0 0 0 0 x 5 x 6 − x 6 − x 5 0 0 − x 3 x 4 x 1 − x ∗ 2 − x 2 − x ∗ 1 x ∗ 4 x ∗ 3 x 4 − x 3 x 2 x ∗ 1 − x 1 x ∗ 2 x ∗ 3 x ∗ 4 0 0 − x ∗ 6 x ∗ 5 − x ∗ 5 x ∗ 6 0 0 x 6 x 5 0 0 0 0 − x 5 − x 6 − x 2 x 1 x 4 x ∗ 3 − x 3 x ∗ 4 − x ∗ 1 − x ∗ 2 x ∗ 1 x ∗ 2 x ∗ 3 x 4 x ∗ 4 − x 3 x 2 − x 1 x ∗ 5 − x ∗ 6 0 0 0 0 x ∗ 6 − x ∗ 5 x 5 x 6 0 0 0 0 − x 6 − x 5 x ∗ 1 x ∗ 2 x ∗ 3 x 4 x ∗ 4 − x 3 x 2 − x 1 − x 2 x 1 x 4 x ∗ 3 − x 3 x ∗ 4 − x ∗ 1 − x ∗ 2 − x ∗ 6 x ∗ 5 0 0 0 0 − x ∗ 5 x ∗ 6 − x 6 x 5 0 0 0 0 x 5 − x 6 − x 2 x 1 − x 4 x ∗ 3 x 3 x ∗ 4 x ∗ 1 x ∗ 2 − x ∗ 1 − x ∗ 2 − x ∗ 3 x 4 − x ∗ 4 − x 3 x 2 − x 1 − x ∗ 5 − x ∗ 6 0 0 0 0 − x ∗ 6 − x ∗ 5 x 5 − x 6 0 0 0 0 − x 6 x 5 x ∗ 1 x ∗ 2 x ∗ 3 − x 4 x ∗ 4 x 3 − x 2 x 1 x 2 − x 1 x 4 − x ∗ 3 − x 3 − x ∗ 4 − x ∗ 1 − x ∗ 2 − x ∗ 6 − x ∗ 5 0 0 0 0 − x ∗ 5 − x ∗ 6 0 0 − x 6 x 5 x 5 − x 6 0 0 − x 4 − x 3 x 2 x ∗ 1 x 1 − x ∗ 2 − x ∗ 3 x ∗ 4 x 3 x 4 − x 1 x ∗ 2 − x 2 − x ∗ 1 − x ∗ 4 x ∗ 3 0 0 − x ∗ 5 − x ∗ 6 − x ∗ 6 − x ∗ 5 0 0 0 0 x 5 − x 6 − x 6 x 5 0 0 − x 3 − x 4 x 1 − x ∗ 2 x 2 x ∗ 1 x ∗ 4 − x ∗ 3 x 4 x 3 − x 2 − x ∗ 1 − x 1 x ∗ 2 x ∗ 3 − x ∗ 4 0 0 − x ∗ 6 − x ∗ 5 x ∗ 5 − x ∗ 6 0 0 0 0 x 5 − x 6 x 6 − x 5 0 0 − x 3 x 4 x 1 − x ∗ 2 − x 2 − x ∗ 1 x ∗ 4 x ∗ 3 − x 4 x 3 − x 2 − x ∗ 1 x 1 − x ∗ 2 − x ∗ 3 − x ∗ 4 0 0 − x ∗ 6 − x ∗ 5 x ∗ 5 − x ∗ 6 0 0 0 0 − x 6 x 5 0 x 6 − x 5 0 x 4 − x 3 x 2 x ∗ 1 − x 1 x ∗ 2 x ∗ 3 x ∗ 4 x 3 − x 4 − x 1 x ∗ 2 x 2 x ∗ 1 − x ∗ 4 − x ∗ 3 0 0 − x ∗ 5 − x ∗ 6 − x ∗ 6 x ∗ 5 0 0 x 5 − x 6 0 0 0 0 x 6 − x 5 x ∗ 1 x ∗ 2 x ∗ 3 x 4 x ∗ 4 − x 3 x 2 − x 1 x 2 − x 1 − x 4 − x ∗ 3 x 3 − x ∗ 4 x ∗ 1 x ∗ 2 − x ∗ 6 − x ∗ 5 0 0 0 0 x ∗ 5 − x ∗ 6 − x 6 x 5 0 0 0 0 − x 5 x 6 − x 2 x 1 x 4 x ∗ 3 − x 3 x ∗ 4 − x ∗ 1 − x ∗ 2 − x ∗ 1 − x ∗ 2 − x ∗ 3 − x 4 − x ∗ 4 x 3 − x 2 x 1 − x ∗ 5 − x ∗ 6 0 0 0 0 − x ∗ 6 x ∗ 5 x 2 x ∗ 1 x 4 − x ∗ 3 − x 3 − x ∗ 4 x 1 − x ∗ 2 − x 6 − x ∗ 5 0 0 0 0 − x ∗ 5 − x 6 x 5 − x 6 ∗ 0 0 0 0 − x 6 ∗ x 5 − x 1 x ∗ 2 x ∗ 3 − x 4 x ∗ 4 x 3 − x 2 − x ∗ 1 x 1 − x ∗ 2 − x ∗ 3 x 4 − x ∗ 4 − x 3 x 2 x ∗ 1 − x ∗ 5 − x 6 0 0 0 0 − x 6 − x ∗ 5 − x 6 ∗ x ∗ 5 0 0 0 0 x 5 − x 6 ∗ − x 2 − x ∗ 1 − x 4 x ∗ 3 x 3 x ∗ 4 − x 1 x ∗ 2 x 4 x 3 − x 2 x 1 x ∗ 1 x ∗ 2 x ∗ 3 − x ∗ 4 0 0 − x 6 − x ∗ 5 − x ∗ 5 − x 6 0 0 0 0 x 5 − x 6 ∗ − x 6 ∗ x 5 0 0 − x 3 − x 4 − x ∗ 1 − x ∗ 2 x 2 − x 1 x ∗ 4 − x ∗ 3 x 3 x 4 x ∗ 1 x ∗ 2 − x 2 x 1 − x ∗ 4 x ∗ 3 0 0 − x ∗ 5 − x 6 − x 6 − x ∗ 5 0 0 0 0 − x 6 ∗ x 5 x 5 − x 6 ∗ 0 0 − x 4 − x 3 x 2 − x 1 − x ∗ 1 − x ∗ 2 − x ∗ 3 x ∗ 4 x 3 − x 4 x ∗ 1 x ∗ 2 x 2 − x 1 − x ∗ 4 − x ∗ 3 0 0 − x ∗ 5 − x 6 x 6 x ∗ 5 0 0 0 0 − x 6 ∗ x 5 − x 5 x 6 ∗ 0 0 x 4 − x 3 x 2 − x 1 x ∗ 1 x ∗ 2 x ∗ 3 x ∗ 4 − x 4 x 3 − x 2 x 1 − x ∗ 1 − x ∗ 2 − x ∗ 3 − x ∗ 4 0 0 − x 6 − x ∗ 5 x ∗ 5 x 6 0 0 0 0 x 5 − x 6 ∗ 0 − x 5 x 6 ∗ 0 − x 3 x 4 − x ∗ 1 − x ∗ 2 − x 2 x 1 x ∗ 4 x ∗ 3 x 1 − x ∗ 2 − x ∗ 3 − x 4 − x ∗ 4 x 3 − x 2 − x ∗ 1 − x ∗ 5 − x 6 0 0 0 0 x 6 x ∗ 5 − x 6 ∗ x 5 0 0 0 0 − x 5 x 6 ∗ − x 2 − x ∗ 1 x 4 x ∗ 3 − x 3 x ∗ 4 x 1 − x ∗ 2 x 2 x ∗ 1 − x 4 − x ∗ 3 x 3 − x ∗ 4 − x 1 x ∗ 2 − x 6 − x ∗ 5 0 0 0 0 x ∗ 5 x 6 x 5 − x 6 ∗ 0 0 0 0 x 6 ∗ − x 5 − x 1 x ∗ 2 x ∗ 3 x 4 x ∗ 4 − x 3 x 2 x ∗ 1 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 5 Fig. 6. The [32 , 32 , 6] code H 5 with fra ction of zeros 1 4 Smarajit Das (S’2007) was born in W est Bengal, India. He completed his B.E. degree at the Sardar V alla bbhai National Institute of T echnol ogy , Surat, India in 2001. He is currently a Ph.D. st udent in the Departmen t of Electric al Communicati on Engineer- ing, Indian Institute of Science, Bangalore, India. His primary research interests incl ude space-time coding for MIMO channels with an emphasis on algebra ic code c onstruction techniqu es. B. Sundar Rajan (S’84-M’91-SM’98) w as born in T a mil Nadu, India. He rece iv ed the B.Sc. degre e in mathematic s from Madras Uni versi ty , Madras, India, the B.T ech de gree in elect ronics fr om Madras Institut e of T e chnology , Madras, and the M.T ech and Ph.D. degre es in electrica l engineering from the Indian Institute of T echnology , Kanpur , India, in 1979, 1982, 1984, and 1989 respecti vel y . He was a faculty member with th e Depa rtment of Electrical Engineeri ng at the Indian Institute of T ec hnology in Delhi, India, from 1990 to 1997. Si nce 1998, he has been a Professor in the Department of Electric al Communica tion Engineering at the Indian Institut e of Science , Bangalore, India. His primary research intere sts are in algebr aic coding, coded modulat ion and space-ti me coding. Dr . Rajan is an Edit or of IEEE Tran sactions on Wirel ess Communication s from 2007 and also a Editorial Board Member of Internationa l Journal of Information and Co ding Theo ry . He is a Fellow of Indi an Na tional Ac ademy of Engin eering and is a Member of the Ameri can Mathemati cal Society .