Golden Space-Time Block Coded Modulation

1 Golden Space-T ime Block Coded Modulation L. Luzzi G. Rekaya-Be n Othman J.-C. Belfiore E. V iterbo Abstract — In this paper we present a block coded modulation scheme fo r a 2 × 2 MIMO system ov er slow fading channels, where the inner code is the Golden Code. Th e scheme is based on a set partiti oning of the Golden Code using two-sided ideals whose norm is a power of two. In this case, a lower bound for the minimum determinant is give n by the minimum Hamming distance. The descript ion of the ring structure of the quotients suggests further optimization in order to improve the overall distribution of determinants. Perf ormance simulations show that the GC-RS schemes achieve a significant ga in ov er the uncoded Golden Code. Index T erms — Golden Code, coding gain, Space-Time Block Codes, Reed-Solomon Codes I . I N T R O D U C T I O N The wide d iffusion of w ireless communica tions has led to a growing dem and f or high -capacity , highly reliable trans- mission sch emes over fading channels. The use of mu ltiple transmit and receive antennas can greatly improve perfo rmance because it increases the diversi ty o r der of th e system, defined as the number of independen t transmit-receive path s. In or der to explo it f ully the a vailable diversity , a ne w cla ss of code designs, c alled Space- T ime Block Codes , was d ev eloped. In the coher ent , bloc k fading mod el, where the chan nel coef- ficients are su pposed to b e k nown at the recei ver, an d remain constant for a time blo ck, the fund amental c riteria for code design are - the rank criterion , stating th at the d ifference of two distinct cod ew o rds or “space-time block s” must b e a full-rank matrix, - the determinant criterion , stating that its minimum d e- terminant ought to be maximized [11]. Codes meetin g these two cr iteria can b e con structed using tools fr om algeb raic numb er th eory . In particular, by cho osing a subset of a division algebra over a n umber field as our co de, we en sure that a ll the non zero codewords ar e invertible. If, further more, this sub set is contained in an or der of the algebra, the minim um d eterminan t over all non zero codewords will b e bound ed f rom below and will n ot vanish when th e size of the constellation grows to infinity . In the 2 × 2 MI MO case, Belfiore et a l. [1] d esigned the Golden Code G , a full-rate, full-r ank and in formatio n-lossless code satisfying the non -vanishing deter minant condition. The n × n MIMO codes that achiev e these proper ties were called P erfect Codes in [8] and also studied in [4]. In this paper we focus on the slow blo ck fadin g cha nnel, where the fading coefficients are assumed to be constant for Jean-Claude Be lfiore, Ghaya Rekaya-Ben Othman and Laura Luzzi are with Ecole Nationale Sup ´ erieure des T ´ el´ ecommunications (ENST), 46 Rue Barrault, 75013 Paris, France. E -mail: { belfiore , rekaya , luzzi } @ e nst . fr . Emanuele V iterbo is with DEIS - Universit ` a Della Calabria, V ia P . Bucci, 42/C, 87036 Rende (CS), Italy . E-mail: viterbo @ deis . unical . it . a certain number of time blocks L . 1 Even th ough fading hinders transmission with respect to the A WGN ca se, fast fadin g is actually bene ficial becau se the transmission paths at different times can be regarded as indepen dent. On the contrar y , with slow fading the ergodicity assumption must be dropp ed and the diversity of the system is reduced, leading to a performance loss. This lo ss can be co mpensated using coded mo dulation : in a general setting , a full-r ank spac e time block code is used as a n inner cod e to guarantee full di versity , and is combined with an outer code which improves the minimum determinant. W e will take as our inner code the Gold en Code: we f ocus o n the pro blem of designing a block cod e { X = ( X 1 , . . . , X L ) } , where each componen t X i is a Golden codew o rd. In order to increa se the minimu m deter minant, one can con- sider the ide als of G . In [6] , Hong et a l. descr ibe a set partitioning of the Golden Code, based on a chain of left ideals G k = G B k , suc h that the minimum determ inant in G k is 2 k times that of G . Choosing the comp onents X i indepen dently in G k , one obtain s a very simple bloc k code. For small sizes o f the signal constellation these subc odes already yield a per forman ce g ain with respect to the “u ncoded ” Gold en Code (that is, with respect to choosing X i ∈ G ind ependen tly). Howe ver , the gain is cancelled out asymptotically by the loss of rate as the size of the signal set grows to infinity , since an energy increase is required to mantain the same spec tral e fficiency , or bit-rate per channel use. A better per forman ce is achieved when the X i are not chosen in an indep endent fashion. In [ 6], two encod ers a re com - bined: a tr ellis enc oder whose o utput belon gs to the quotient G k / G k +1 , and a lattice encoder for G k +1 ( T r ellis Coded Modulation ). The global min imum deter minant for the block code is given by ∆ min = min X 6 =0 det L X i =1 X i X H i ! This expression is difficult to handle because its “mixed terms” are Frobeniu s norm s of p roducts in G . The codes described in [6] are design ed to maxim ize the appr oximate parameter ∆ ′ min = min X 6 =0 P L i =1 det  X i X H i  and so a priori they might b e su boptimal; we will h ere con sider th e m ixed terms and so obtain a tighter bound for ∆ min . A rou gh estimate of th e cod ing gain for the block co de co mes from its minimum “Hamm ing distance”, th at is, the minimu m 1 This kind of behaviour might be cause d by large obstructio ns between transmitt er and recei ver . The model is realistic if L is smaller than the coheren ce time of the channel; for most practic al applicatio ns, it has been estimate d [2] tha t the coherence time is greater than 0 . 01 seco nds, so th at L < 100 is a legitima te assumption. Submitted to IEEE T ra ns. on Inform. The ory , Dec. 2007 number of n onzero comp onents. T o increa se th e Hamm ing weight, we will take as our ou ter co de an error corr ecting code over the quotien t of G by one of its ideals. The ch oice of the id eal must follo w som e basic re quiremen ts. First of all, in order to d o a binary partitioning , we need to choose ideals who se index is a power of 2 . Moreover, we will choose two -sided ideals to ensure that the qu otient grou p is also a ring. W e will describe the ideals of G th at satisfy our r equirem ents; in particular, we consider th e quotient rings G / (1 + i ) G and G / 2 G , which turn ou t to be iso morph ic to the rings of 2 × 2 matrices over F 2 and F 2 [ i ] respectively . Unfortu nately , little is known about codes over non- commutative rin gs, and for the time being we ha ve been un able to exploit the ring stru cture directly for cod e con struction, except in the simple case of the r ep etition code over the cosets of (1 + i ) G . Our perform ance simulatio ns show th at this basic construction can lead to up to 2 . 9 dB of gain with respect to the “uncod ed” case. From th e additive p oint of vie w the qu otient G / 2 G is indis- tinguishable from F 256 , for which a wid e variety o f error- correcting c odes are a vailable. W e can co mbine a shorten ed Reed-Solomo n code with the encod er of the q uotient ring to increase the minimum Hamming distance of the code. Simulation resu lts show tha t using 4 -QAM constellations, that is using only one lattice point per coset, and with codes of length L = 4 and L = 6 , we o btain a gain of 6 . 1 dB and 7 . 0 dB with re spect to the uncod ed Gold en Code a t the same spectral effi ciency . The co nstruction ca n be extended to th e case of 16 -QAM modulatio n with multiple points p er coset, wh ere the ga in is somewhat smaller ( 3 . 9 dB f or L = 4 ), being limited by the minimum distance in the ideal. The p aper is organize d as follows: in Sec tion I I, we re- call the algeb raic c onstruction of the Golden Cod e and its proper ties. I n Section III, we describe the general setting for Golden block code s an d the codin g ga in estimates; in Section IV, we study the “good ideals” of G for binary partitioning . In Sections V and VI we introduce th e repetitio n code and the Reed-So lomon b lock code over G and discuss their performan ce obtained through simulations. The interested reader can fin d in the Appe ndix the ma in defin itions and theorems concern ing q uaternio n algeb ras that are cited in the paper . I I . T H E G O L D E N C O D E Since we are interested in the partition ing of th e Go lden Code, we begin by r ecalling its algebr aic construc tion. For the sake of simplicity , definitio ns and theorem st atements are collected in the Appendix. The Go lden Cod e G , intro duced in [1] , is optimal for th e case o f 2 transmit and 2 or m ore r eceiv e a ntennas. This code is constructed using the cyclic division alg ebra A = ( Q ( i, θ ) / Q ( i ) , σ, γ ) o ver th e num ber field Q ( i, θ ) , wher e θ = √ 5+1 2 is the golden number . The set A is th e Q ( i, θ ) -vector space Q ( i, θ ) ⊕ Q ( i, θ ) j , where j is such that j 2 = γ ∈ Q ( i ) ∗ , xj = j ¯ x ∀ x ∈ Q ( i, θ ) . Here we de note b y σ th e canonical c onjugacy sending an element x = a + b θ ∈ Q ( i, θ ) to ¯ x = a + b ¯ θ , where ¯ θ = 1 − θ = 1 − √ 5 2 , θ ¯ θ = − 1 As its degree over its center Q ( i ) is 4 , A is also called a quaternio n algebra . If we choose γ = i , γ is not a norm in Q ( i, θ ) / Q ( i ) [1], and this implies that A is a division algebr a (see Th eorem 8 in the Append ix). From Theo rem 9, it follows that Q ( i, θ ) is a splitting field for A , and so A is isom orphic to a subalgeb ra of M 2 ( Q ( i, θ )) . The inclusion is gi ven by x 7→  x 0 0 ¯ x  , ∀ x ∈ Q ( i , θ ) , j 7→  0 1 i 0  (1) That is, e very element X ∈ A admits a matrix representation X =  x 1 x 2 i ¯ x 2 ¯ x 1  , x 1 , x 2 ∈ Q ( i, θ ) (2) The Golden Cod e G is a subr ing of A h aving two additional proper ties: the minimu m determinant δ = min X 6 = X ′ , X,X ′ ∈G | det( X − X ′ ) | 2 should be strictly b ound ed away fro m 0 , and mor eover we want the code to be in formation lossless. For the first c ondition , if we r equire that the matrix elements of X belo ng to the ring o f integers Z [ i, θ ] of Q ( i, θ ) , th en X belongs to the Z [ i ] -or der O =  x 1 x 2 i ¯ x 2 ¯ x 1  , x 1 , x 2 ∈ Z [ i, θ ]  (3) Since x ∈ Z [ i, θ ] implies that the reduced n orm N ( x ) = x ¯ x belongs to Z [ i ] , we hav e det( X ) ∈ Z [ i ] , so | det( X ) | ≥ 1 for ev e ry X ∈ O \ { 0 } . Each codeword of O c arries two symbols x 1 = a + bθ , x 2 = c + dθ in Z [ i, θ ] , or equivalently four informa tion symbols ( a, b, c, d ) ∈ Z [ i ] 4 : the code is full-rate . In ord er to have an info rmation lo ssless cod e, a right p rincipal ideal of O of the form α O was u sed, whe re α = 1 + i ¯ θ : its matrix representation is A =  α 0 0 ¯ α  ∈ O (4) The Golden Code is defined as G = 1 √ 5 α O . Every code word in G is of the form X = 1 √ 5 AW , with W ∈ O : X = 1 √ 5  α ( a + bθ ) α ( c + dθ ) ¯ αi ( c + d ¯ θ ) ¯ α ( a + b ¯ θ )  (5) Remark 1. W e hav e seen that ∀ W ∈ O \ { 0 } , | det( W ) | ≥ 1 . Consequently , ∀ X ∈ G \ { 0 } , | det( X ) | 2 ≥ δ = 1 5 . In fact, if X = A √ 5 W , | det( X ) | = | N ( α ) | 5 | det( W ) | =    det ( W ) √ 5    , since | N ( α ) | = | 2 + i | = √ 5 . The code G has cubic shapin g : it is isom etric to the c ubic lattice Z [ i ] 4 (and so it is infor mation lossless). In fact, if we 2 Submitted to IEEE T ra ns. on Inform. The ory , Dec. 2007 consider the lin ear mapping φ : A → C 4 that vectorizes matrices φ  a c b d  = ( a, b, c, d ) ∈ C 4 , then φ ( G ) = R Z [ i ] 4 , where R is the unitary matrix R = 1 √ 5     α − ¯ αi 0 0 0 0 ¯ αi α 0 0 α − ¯ αi ¯ α − αi 0 0     (6) Even though G is d efined as a right ideal, it is easy to see that actually it is a two-sided ideal : if w = w 1 + w 2 j ∈ O , w 1 , w 2 ∈ Z [ i, θ ] , α ( w 1 + w 2 j ) = w 1 α + w 2 j ¯ α = ( w 1 + iθ w 2 j ) α, observing that αiθ = iθ + 1 = ¯ α . But ξ : w 1 + w 2 j 7→ w 1 + iθw 2 j (7) is an homomo rphism of Z [ i ] -mo dules that map s O into itself bijectively , therefore α O = O α . Finally , √ 5 G is an inte gral ideal because it is c ontained in O . Remark 2. For the sake of simplicity , in this section we h ave described the Golden Code as an infinite code. Ho wever in a practical transmission scheme, one consider s a fin ite subset of G , by choosing the infor mation symbols a, b, c, d in a QAM constellation carved fr om Z [ i ] . I I I . G O L D E N B L O C K C O D E S W e now focus on the case of a slow blo ck fadin g channel, meaning th at th e chan nel coefficients remain constant during the transmission of L co dew ords. The transmitted signal X = ( X 1 , . . . , X L ) will be a vector of Golden codewords in a b lock code S ⊂ G L . The receiv e d si gnal is given by Y = H X + W , X , Y , W ∈ C 2 × 2 L , (8) where th e entries of H ∈ C 2 × 2 are i.i.d. comp lex Gaussian random variables with zero me an an d variance per real d imen- sion equal to 1 2 , and W is th e co mplex Gaussian noise w ith i.i.d. en tries o f zero mean and variance N 0 . W e co nsider th e coheren t case, where the chan nel matrix H is known at the receiver . The pairwise error probability is bounded by [11] P ( X 7→ X ′ ) ≤ 1  √ ∆ min E S N 0  4 , (9) In th e above f ormula, E S is the average energy per symbol of S and ∆ min = min X ∈S \{ 0 }   det( XX H )   In ord er to minimize the PEP fo r a given SNR, we should maximize ∆ min . W e will show th at   det( XX H )   ≥ ( w H ( X )) 2 δ, where w H ( X ) is the number of nonzero codewords in ( X 1 , . . . , X L ) (a sort of “Hamming weight”), and δ = 1 5 is the minimum square determinant of the Golden Cod e. Because of the lac k of d iv ersity of the chan nel in the slow fading case, if we simply choo se X 1 , . . . , X L indepen dently in the Golden Code, the code performan ce will be poor compar ed to the f a st block fading model. W e call this schem e th e “uncod ed Golden Code ”: in this case ∆ min = δ , f or any length L . T o compare th e erro r pro bability of a blo ck codes with that of the u ncoded Golden Code of equal le ngth L with the same data rate, we can employ the asymptotic coding gain de fined in [6]: γ as = √ ∆ min /E S p ∆ min ,U /E S ,U , (10) where ∆ min , ∆ min ,U and E S , E S ,U are the minimum deter- minants and average constellation ene rgies of the block cod e and the uncoded case respectiv ely . In all the cases th at we considered , the th eoretical gain γ as turned out to be smaller th an the actual ga in evidenced by computer simulations. This is no t su rprising, since γ as is only a compariso n of the do minant terms in the pairwise erro r probab ility . A. Estimates of the F r obeniu s norm First of all, w e g iv e a m ore exp licit expression fo r det( XX H ) . W e define the quatern ionic co njugacy in the algeb ra A : X =  x 1 x 2 i ¯ x 2 ¯ x 1  7→ e X =  ¯ x 1 − x 2 − i ¯ x 2 x 1  Observe th at ∀ X ∈ A , e X X = det( X ) 1 (11) e X + X = ( x 1 + ¯ x 1 ) 1 = tr( X ) 1 (12) det( X ) = det( e X ) (13) where 1 denotes the identity m atrix. Recall that the F r obenius norm of a matrix M = ( m i,j ) is k M k F = s X i,j | m i,j | 2 Then the following for mula holds: Lemma 1 . ∀ X = ( X 1 , . . . , X L ) ∈ A L , det( XX H ) = det L X i =1 X i X H i ! = = | det( X 1 ) | 2 + . . . + | det( X L ) | 2 + X j >i    e X j X i    2 F (14) The proof can be found in Appendix I. W e also st ate some simple prop erties of the quaternionic conjuga te an d of the Froben ius norm that will be usef ul in the sequel: Remark 3. a) If W ∈ O , k W k 2 F ∈ Z . b) Let X, Y be tw o 2 × 2 com plex-valued matrices. Then k X k 2 F ≥ 2 | det( X ) | ,    e X Y    2 F ≥ 2 | det( X ) | | det( Y ) | (15) 3 Submitted to IEEE T ra ns. on Inform. The ory , Dec. 2007 In particular ∀ W ∈ O \ { 0 } , k W k 2 F ≥ 2 | det( W ) | ≥ 2 (16) c) If X 1 , X 2 ∈ G \ { 0 } ,    e X 2 X 1    2 F ≥ 2 5 = 2 δ (17) From equation (1 5), it fo llows that the deter minant is bound ed fr om belo w by the squared Ha mming weight: Lemma 2 . Let X = ( X 1 , . . . , X L ) ∈ G L . Then det( XX H ) ≥ L X i =1 | det( X i ) | ! 2 ≥ ( w H ( X )) 2 δ, wher e w H ( X ) = # { i ∈ { 1 , . . . , L } | X i 6 = 0 } is the Ha mming weight of the block X . I V . T W O - S I D E D I D E A L S O F G The ch oice of a good block cod e of length L will be b ased on a partition chain of ideals of the Golden Code. W e would like to ob tain a bin ary p artition, wh ich is simpler to use for codin g and fu lly compatible with the choice of a QAM constellation: we must then use ideals whose ind ex is a power of 2 , that is, whose n orm is a power of 1 + i . A similar co nstruction ap pears in [6] and employs one- sided ideals. Howe ver, in ord er to have goo d estimates of the cod ing gain, beca use of the mixed terms in the m inimum d eterminan t formu la (14), we need to take the ring structu re into account: we will ch oose two- sided ideals to ensure that the id eals are in variant with respect to the quaternio nic conjug acy an d multiplication on both sid es, and that the quo tient gro up is also a ring. In this section we describe the structur e o f the two-sided ideals o f G whose no rm is a power of 1 + i . Unfortunately , we will see that the only two-sided ideals with this property are the tri vial ones. W e then study the co rrespon ding qu otient rings, which are rings of matrices o ver n on-integral rings. For these constructions we will need some notions fro m non - commutative algebra (see Ap pendix II I), relating the existence of two-sided ideals to the ramification of primes over th e base field. W e will also show that O is a maximal order of A . As we ha ve seen in Section II, O = Z [ i, θ ] ⊕ Z [ i, θ ] j is a Z [ i ] -order of A , a nd G = √ 5 G = α O is a tw o-sided principa l ideal of O . √ 5 G is also a prime ideal since √ 5 G ∩ Z [ i ] = (2 + i ) is a prime ideal of Z [ i ] (see Theorem 12 in the App endix) . Observe that the prime ideals (2 + i ) an d (2 − i ) of Z [ i ] are both ramified in A : in f act (2 + i ) = ( α ) 2 , and (2 − i ) = ( α ′ ) 2 , where α ′ = 1 − i ¯ θ (Remark that α = i θ ¯ α , α ′ = − i ¯ θ ¯ α ′ ). Proposition 3. O is a maximal or der . Pr oof. A is a qu aternion algebr a u nramified at infin ity: the infinite prime s are co mplex (be cause the b ase field Q ( i ) is imaginary quadra tic) an d they can ’t be r amified. Then one can ch eck that O is max imal throug h the comp utation of its reduced discriminant d ( O ) (see Pro position 15 in the Append ix). d ( O ) is eq ual to p | det(tr( w k w l )) | Z [ i ] , where { w 1 = 1 , w 2 = θ , w 3 = j , w 4 = θ j } is the basis of O over Z [ i ] : ( w k w l ) 1 ≤ k,l ≤ 4 =     1 θ j θ j θ θ 2 θj θ 2 j j ¯ θj i i ¯ θ θj − j θi − i     , det(tr( w i w j )) = det     2 1 0 0 1 3 0 0 0 0 2 i i 0 0 i − 2 i     = 25 Then d ( O ) = 5 Z [ i ] . I f O were strictly co ntained in a maximal order O ′ , d ( O ′ ) would be strictly larger than 5 Z [ i ] . But we kn ow from Pro position 1 5 that d ( O ′ ) is the p rodu ct of all ramified primes of A ; in par ticular it should be contain ed in the ideals (2 + i ) and (2 − i ) . But then it would b e con tained in 5 Z [ i ] , a contradiction . Then O is a maximal order, and G is a normal ideal. Since O is maximal, f rom Proposition 15 we also learn that (2 + i ) and (2 − i ) are the only ramified primes in A . Then T heorem 1 6 implies tha t the p rime two-sided ideals of O a re eith er of the f orm p O , where p is prime in Z [ i ] , or belong to { α O , α ′ O} . It follows that the o nly two-sided ideals o f G whose n orm is a power of 1 + i are the tri v ial ideals of th e form (1 + i ) k G . A. The quotien t ring G / (1 + i ) G In the sequel, we will denote by G the integral ideal √ 5 G . Consider the p rime idea l (1 + i ) O . G and (1 + i ) O are coprime ideals, that is G + (1 + i ) O = O ; as a conseque nce, G ∩ (1 + i ) O = G (1 + i ) O = (1 + i ) G . Recall the following basic result: Theorem 4 (third isomo rphism theorem for rings) . Let I and J be ideals in a ring R . Then I I ∩ J ∼ = I + J J . If I = G and J = (1 + i ) O , we get G (1 + i ) G ∼ = O (1 + i ) O (18) If π G : G → G / (1 + i ) G and π O : O → O / (1 + i ) O are the canonical pro jections o n th e quotien t, the ring iso morph ism in (18) is simply gi ven by π G ( g ) 7→ π O ( g ) . Theorem 12 imp lies th at O / (1 + i ) O is a simple algebra over Z [ i ] / (1 + i ) ∼ = F 2 . W e d enote th e image of x ∈ O throu gh π O with [ x ] . Lemma 5. O / (1 + i ) O is isomorphic to the ring M 2 ( F 2 ) of 2 × 2 m atrices over F 2 . Pr oof. W e use the well-known lemma [7]: Lemma 6. Let R be a ring with identity , I a pr o per ideal o f R , M a fr ee R -mod ule with basis X a nd π : M → M / I M the canon ical pr ojection. Then M /I M is a fr ee R/ I -module with basis π ( X ) and | π ( X ) | = | X | . 4 Submitted to IEEE T ra ns. on Inform. The ory , Dec. 2007 W e know that O / (1 + i ) O is a Z [ i ] -mod ule; the lem ma implies that it is also a free Z [ i ] / (1 + i ) -mod ule, tha t is a vector s pace over F 2 , whose basis is { [1] , [ θ ] , [ j ] , [ θ j ] } . W e de fine an hom omorp hism of F 2 -vector spaces ψ : O / (1 + i ) O → M 2 ( F 2 ) by specifying the images of the basis: ψ ([1]) = 1 , ψ ([ θ ]) =  0 1 1 1  , ψ ([ j ]) =  0 1 1 0  , ψ ([ θ j ]) = ψ ([ θ ]) ψ ([ j ]) It is one- to-one since ψ ([1]) , ψ ([ θ ]) , ψ ([ j ]) , ψ ([ θ j ]) are linearly indepen dent. T o prove tha t ψ is also a ring homomor phism, it is sufficient to verify that ψ ( w i w j ) = ψ ( w i ) ψ ( w j ) for all pairs of basis vectors w i , w j . Recall th at as a Z [ i ] -lattice, G is isom etric to √ 5 Z [ i ] 4 , and a canonical basis is given by { α, αθ , αj, αθ j } . The correspo nding elements ψ ([ α ]) , ψ ([ αθ ]) , ψ ([ αj ]) , ψ [ αθ j ]) of M 2 ( F 2 ) are e 1 =  0 1 1 1  , e 2 =  1 1 1 0  , e 3 =  1 0 1 1  , e 4 =  1 1 0 1  . (19) It is easy to chec k that the only in vertible elements in M 2 ( F 2 ) are e 1 , e 2 , e 3 , e 4 , e 1 + e 2 = 1 , e 3 + e 4 = ϕ ( j ) Observe that the lifts to G o f no n-invertible elements have a higher determinant: Remark 4. I f M ∈ M 2 ( F 2 ) \ { 0 } is non-in vertible, min X ∈G , π G ( √ 5 X )= M | det( X ) | 2 ≥ 2 δ Pr oof. π G ( X ) is non-invertible in G / (1 + i ) G if and on ly if its determinan t is non-invertible i n Z [ i ] / (1 + i ) , that is, det( X ) = e X X ∈ (1 + i ) \ { 0 } . (If M 6 = 0 , det( X ) 6 = 0 , since A is a division r ing.) Then    det( e X X )    = | det ( X ) | 2 ≥ 2 δ . B. The qu otient ring G / 2 G Again, G and 2 O are cop rime and so G + 2 O = O , G ∩ O = 2 G ; from the third isomo rphism th eorem for rings, G 2 G ∼ = O 2 O . Lemma 7. O / 2 O is isomorphic to the ring M 2 ( F 2 [ i ]) of 2 × 2 matrices over the rin g F 2 [ i ] . Pr oof. First of all, Lemma 6 im plies that O / 2 O is a free Z [ i ] / 2 -mo dule, that is a free F 2 [ i ] -mod ule, of d imension 4 . As in the previous case, we ca n construct an explicit homom orphism o f F 2 [ i ] -mod ules φ : O / 2 O → M 2 ( F 2 [ i ]) : φ ([1]) = 1 , φ ([ θ ]) =  1 + i 1 i i  , φ ([ j ]) =  0 1 i 0  , φ ([ θj ]) = φ ([ θ ]) φ ([ j ]) One can easily check that φ is bijective (the images o f the basis elemen ts being linear ly indepen dent) a nd that it is a ring homom orphism. T o find an explicit isomor phism between G / 2 G and M 2 ( F 2 ) , consider the following d iagram, where π G : G → G / 2 G is the projection on the quotient, ϕ is given by the third isomorph ism th eorem for rings, and φ : O / 2 O → M 2 ( F 2 [ i ]) is the mapping defined in L emma 7: G π G − − − − − → G / 2 G ϕ − − − − → O / 2 O φ − − − − → M 2 ( F 2 [ i ]) The basis { α, αθ , αj, αθ j } of G as a Z [ i ] -mo dule is also a basis of G / 2 G as an F 2 [ i ] -mod ule. The isomorp hism ϕ is simply the composition of the inclusion G ֒ → O and the quotient mod (1 + i ) O . W e can com pute the images thr ough φ of the basis vectors: ob serving that α = 1 + i − i θ , αθ = θ − i , αj = (1 + i − iθ ) j, αθ j = ( θ − i ) j, we get φ ( α ) =  0 i 1 i  , φ ( αθ ) =  1 1 i 0  , (20) φ ( αj ) =  1 0 1 1  , φ ( αθj ) =  i 1 0 i  . (21) Also in this case, the lifts X of non -inv e rtible elements of M 2 ( F 2 [ i ]) in G will ha ve non-invertible determinant, that is | det ( X ) | 2 ≥ 2 . C. The encod er The codes tha t we consid er follow th e gener al outline of Forney’ s coset c odes , tak ing advantage of the de composition G = [ G /I ] + I , where I is (1 + i ) G or 2 G , and [ G /I ] deno tes a set of coset leaders. - a binary ( n, k , d min ) encoder operates o n some of the informa tion d ata, and these coded bits are used to select ( C 1 , . . . , C L ) ∈ ( G /I ) L . - the r emaining info rmation bits are left unco ded and used to select ( Z 1 , . . . , Z L ) ∈ I L . - the correspond ing b lock codeword is X = ( c 1 + Z 1 , . . . , c L + Z L ) ∈ G L , where c i is the coset lead er of C i . The encoder is illustrated in Figure 1 . For a coset code, ∆ min is bounded by the minimum de termi- nant o f I and the minimum distan ce d min of the binary code: ∆ min ≥ min  min X ∈ I \{ 0 } | det( X ) | 2 , d 2 min δ  (22) In fact, if ( c 1 , . . . , c L ) = 0 , then X ∈ I L , and for X 6 = 0 , det( XX H ) ≥ min X ∈ I \{ 0 } | det( X ) | 2 . If on the con trary ( c 1 , . . . , c L ) 6 = 0 , ther e are at lea st d min compon ents of X which do not belong to I , and consequen tly are nonzero, and det( XX H ) ≥ δ w H ( X ) ≥ δ d 2 min . So the performan ce of a coset code will be always limited by the minimum d eterminant of I , excep t if the code on I L is the zero code. If I is simply (1 + i ) G or 2 G , the set of possible co ordinate s 5 Submitted to IEEE T ra ns. on Inform. The ory , Dec. 2007 k 1 bits   k 2 bits   binary encoder   quotien t ring G /I / / L   ideal I o o G Fig. 1. T he general structure of the encoder . ( a, b, c, d ) f or the coset leade rs o f I in G coincides with the (BPSK) 4 and ( 4 -QAM) 4 constellations respectively . This makes it much easier to implem ent co set co des with h igh Hamming distance. V . T H E R E P E T I T I O N C O D E Here we consider the case wh ere I = (1 + i ) G , and th e binary c ode is simply the repetition code of length 2 over G /I . If π : G → G / (1 + i ) G is the projection on the quotient ring ( π ( X ) = π G ( √ 5 X ) ), we de fine C = { X = ( X 1 , X 2 ) ∈ G 2 | π ( X 1 ) = π ( X 2 ) } A. The minimu m determinant Recall that as we ha ve seen in Lemma 1, det( XX H ) = | det( X 1 ) | 2 + | det ( X 2 ) | 2 +    e X 2 X 1    2 F W ith the code C , we hav e ∆ min = 4 δ . In fact if ( X 1 , 0) (respectively , (0 , X 2 ) ) is a codeword o f Hamm ing weight 1 , clearly π ( X 1 ) = 0 and det( XX H ) = | det( X 1 ) | 2 is greater than the minimum squa re determina nt in (1 + i ) G , which is 4 δ . If on the contrary π ( X 1 ) = π ( X 2 ) 6 = 0 , det( XX H ) ≥ ( | det( X 1 ) | + | det( X 2 ) | ) 2 ≥ 4 δ because of equation (15). By choosing any b ijection h of the quotien t rin g G / (1 + i ) G in itself, one ob tains a simple variation o f the rep etition sch eme: C h = { X = ( X 1 , X 2 ) ∈ G 2 | π ( X 2 ) = h ( π ( X 1 )) } Remark 5. A suitable choice of h can slightly improve perfor mance. In th e case of the repetition co de, sup pose th at π ( X 1 ) = π ( X 2 ) = C i . - If C i is invertible in M 2 ( F 2 ) , th en e C i C i = det( C i ) 1 = 1 = e 1 + e 2 in the basis (1 9), and so the min imum determinan t of a codeword e X 2 X 1 ∈ π − 1 ( e C i C i ) is also 1 , an d the m inimum of    e X 2 X 1    2 F is 2 δ . Thu s det( XX H ) ≥ (1 + 1 + 2) δ = 4 δ . - If on the other side C i correspo nds to a non-invertible, nonzer o elemen t in M 2 ( F 2 ) , then (see Remark 4) min X ∈ π − 1 ( C i ) | det( X ) | ≥ √ 2 δ and det( XX H ) ≥ ( | det( X 1 ) | + | det( X 2 ) | ) 2 ≥ (2 √ 2 δ ) 2 = 8 δ . This remark suggests that it might b e m ore conv enient to consider a group ho momor phism h : M 2 ( F 2 ) → M 2 ( F 2 ) which maps in vertib le elements into non -in vertible elements, raising th e minimum d eterminan t to 6 δ if C i in vertib le, h ( C i ) non-invertible:    e X 2 X 1    2 F ≥ 2 √ 2 δ , but    e X 2 X 1    2 F ∈ δ Z (see Remark 3) and so    e X 2 X 1    2 F ≥ 3 δ , and det( XX H ) ≥ (1 + 2 + 3) δ = 6 δ . Such a function ¯ h is not dif ficu lt to define, and in the case of 4 − QAM modulation , an exhaustive search on the finite lattice shows that th e d istribution of determinan ts for C ¯ h is indeed better . 2 B. The encod er Only 4 b its ar e needed to select an element of G / (1 + i ) G ∼ = M 2 ( F 2 ) , while the numb er of bits need ed to select an element in the ideal depen ds on th e chosen modulatio n scheme. Using 4 -QAM constellation s, the two cho ices of an elem ent in (1 + i ) G requ ire 4 bits each: in total, eac h codeword car ries 12 informatio n bits, yieldin g a spectral efficiency o f 3 bpcu. Suppose that ( b 1 , . . . , b 12 ) is the binary input: - ( b 1 , . . . , b 4 ) are u sed to select the matrix b 1 e 1 + b 2 e 2 + b 3 e 3 + b 4 e 4 ∈ M 2 ( F 2 ) in th e ba sis ( 19). T he corre- sponding elem ent of [ G / (1 + i ) G ] is C = [ b 1 α + b 2 αθ + b 3 αj + b 4 αθj ] . - ( b 5 , . . . , b 12 are used to select two codewords i n (1 + i ) G : X 1 = (1 + i )( b 5 α + b 6 αθ + b 7 αj + b 8 αθj ) , X 2 = (1 + i )( b 9 α + b 10 αθ + b 11 αj + b 12 αθj ) . - The final block codeword is ( C + X 1 , h ( C ) + X 2 ) . C. Asymptotic coding gain Since th e min imum d eterminant do esn’t change, the asy mp- totic coding gain estimate is the same for all choices of h . W e c ompare these schemes with the uncod ed Golden Code at 3 bpcu, u sing 4 -QAM co nstellations for the symbols a, c and BPSK co nstellations for the symbo ls b, d in each Golde n codeword (see equation 5). T he average energy per symbol is E S = 0 . 5(0 . 5 + 0 . 25) = 0 . 3 75 , and γ as = √ ∆ min /E S p ∆ min ,U /E S ,U = 2 / 0 . 5 1 / 0 . 37 5 = 1 . 5 , This com putation giv es a theor etical gain of at least 10 log 10 (1 . 5) dB = 1 . 7 dB . 2 In fact, if we define ¯ h ( e 1 ) = e 1 + e 2 + e 4 , ¯ h ( e 2 ) = e 2 + e 3 + e 4 , ¯ h ( e 3 ) = e 1 + e 2 + e 3 , ¯ h ( e 4 ) = e 1 + e 3 + e 4 with respect to the basis (19), we hav e X X ∈C q Det ( XX H ) = 1 + 66 q 4 + 120 q 8 + 48 q 10 + 202 q 16 + . . . X X ∈C ¯ h q Det ( XX H ) = 1 + 24 q 4 + 61 q 8 + 24 q 9 + 8 q 10 + 74 q 12 + . . . 6 Submitted to IEEE T ra ns. on Inform. The ory , Dec. 2007 6 8 10 12 14 16 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 SNR FER Uncoded Golden Code at 3bpcu, channel constant for 2 blocks Repetition code Variation on the repetition code Fig. 2. Performance of the repetition code C Id and of the variat ion C ¯ h at 3 bpcu compared wit h the uncoded Golden Code scheme with the same spec tral ef ficiency . The channel is supposed to be constant for 2 time blocks. Simulation r esults Figure 2 shows the perfor mance of th e codes C Id and C ¯ h , which g ain 2 . 4 dB and 2 . 9 dB respectiv ely over the uncoded scheme at 3 bp cu at the fr ame err or rate o f 10 − 3 , suppo sing that the channel is constant for 2 tim e blocks. V I . G O L D E N R E E D - S O L O M O N C O D E S The re petition code has the ad vantage of simplicity , but clearly its per forman ce is limited by the fact that the mini- mum Hamming distance is only 1 . T o increase the Hamming distance, we need to use a more sophisticated error-corre cting code. As we have seen in th e previous sectio ns, in addition to th e minimum Ham ming distance, also the multiplica tiv e struc ture and the minimum number of non-invertible comp onents h ave a significant influen ce on the co ding ga in of a blo ck c ode design. Id eally , in order to keep track of th ese par ameters, one oug ht to em ploy err or-correcting cod es on M 2 ( F 2 [ i ]) . Howe ver, at pre sent very little is known abou t codes over non- commutative rings; we choose sho rtened Reed-So lomon cod es instead because they are maxim um distance sep arable and t heir implementatio n is very simple ; we will restrict o ur attentio n to the additiv e structure, defining a group isomorphism between G / 2 G and the finite field F 256 . A. The 4- QAM case Using 4 - QAM constellation s to modu late each of th e 4 symbols a, b, c, d in a Golden codew o rd ( 5), we obtain a total of 256 codewords, one in each coset of 2 G . W e con sider an ( n, k , d min ) Reed -Solomon code over F 256 . Each qu adruple ( a, b, c, d ) of 4 -QAM signals carr ies 8 b its or one by te; each block of n Golden cod ew o rds will carry n bytes, correspon ding to k infor mation bytes. The encoding procedu re inv o lves several steps: a) Reed-So lomon en coding: Each info rmation by te can be seen a s a binary p olyno mial of degre e ≤ 8 , that is, an elemen t of the Galo is Field F 256 . An inform ation message o f k bytes, seen as a vector U = ( U 1 , . . . , U k ) ∈ F k 256 , is encode d into a codeword V = ( V 1 , . . . , V n ) ∈ F n 256 using the RS ( n, k , d min ) shortene d code C . For o ur pur poses, it is much better to use a systematic version of the code th at p reserves th e first k bits of the inpu t. b) F r o m the Galois field F 256 to the matrix ring M 2 ( F 2 [ i ]) : W e can repre sent the elemen ts of M 2 ( F 2 [ i ]) as bytes, simply by vectorising each matrix and separating real and im aginary parts. Since we are only work ing with the additive structure, we can identify F 256 and M 2 ( F 2 [ i ]) , which are both F 2 -vector spaces of dimension 8 . According to our simu lation results, it seems that the choice of th e linear identification has very little influence on the code performan ce. c) F r om the matrix ring M 2 ( F 2 [ i ]) to the quotien t ring G / 2 G : For this step we make u se of the isomorphism of F 2 [ i ] - modules ( ϕ ◦ φ ) − 1 : M 2 ( F 2 [ i ]) → G / 2 G d escribed in Section IV -Bthat relate s the coordinates with respect to the bases B G = { α, αθ , αj, αθ j } and (20). Let ( a, b, c, d ) ∈ Z 2 [ i ] 4 be the coordinates of a code word in the basis B G . d) Golden Code encoding : For each o f the n vector componen ts, the symb ols a , b , c , d ∈ Z 2 [ i ] correspond to four 4 -Q AM sign als, a nd can be encoded into a Golden codew ord of the f orm (5). Thus we ha ve obtained a Golde n block X = ( X 1 , X 2 , . . . , X n ) = ξ ( V ) , where ξ : F n 256 → G n is injecti ve. B. Deco ding ML dec oding consists in th e search for the minimum o f the Euclidean distance n X i =1 k H X i − Y i k 2 over all the im ages X = ξ ( V ′ ) of Reed -Solomo n codewords. One can first co mpute and store in m emory the Euclid ean distances d ( i, j ) =    H X ( j ) − Y i    2 (23) for every com ponen t i = 1 , . . . , n of the received vector Y and fo r all th e G olden cod ew o rds X ( j ) , j = 0 , .., 2 55 that can be obtained from a quadruple U ( j ) of 4 -QAM symbols. The searc h fo r the min imum can be carried out using the V iterbi algorithm or a tr ee search algorithm. 1) Stack decod ing: For our co mputer simulation s, we h av e chosen to use a stack decoding alg orithm. I f the code is based on an ( n, k , d min ) Reed-Solomon co de with systematic generato r matrix, th e (256) k codewords ar e the po ssible p aths in a full tree with heig ht k an d 25 6 outg oing bra nches per node. The de coder will store in a stack a cer tain nu mber of triples ( s, u , d u ) , wh ere u is an incomplete path of length s in the tree, and d u is its distance fro m the initial segment ( Y 1 , . . . , Y s ) of Y . An upper bound T for the minimum distance of the receiv ed 7 Submitted to IEEE T ra ns. on Inform. The ory , Dec. 2007 point to the lattice of Golden- RS cod ew o rds will b e used as a “cost function ” for the stack. a) Sorting of distances: Before the sear ch, for each com- ponen t i , the d istances { d ( i, j ) } j =0 ,.., 255 of equation (23) are sorted in increasing order: let d ( i, j 1 ( i )) , d ( i, j 2 ( i )) , . . . , d ( i, j 256 ( i )) be the resulting sequence. b) F irst step: At the beginning, the initial segments of length 1 are inserted into a pre v iously empty stack : the triples (1 , j 1 (0) , d (0 , j 1 (0))) , . . . , (1 , j 256 (0) , d (0 , j 256 (0))) are entered in decreasing o rder w ith r espect to the distance, discarding those whose distances are greater than T . c) I ntermediate step s: At each iteration of the algorith m, the triple ( s, u = ( j (1) , . . . , j ( s ) ) , d u ) curren tly at th e top of the stack is examined. • If s < k , its “children ” nodes ( s, ( u , r ) = ( j (1) , . . . , j ( s ) , r ) , d ( u ,r ) ) , for r = j 1 ( s + 1) , j 2 ( s + 1) , . . . , j 256 ( s + 1) are generated , u pdating the corre spondin g Eu clidean dis- tances: d ( u ,r ) = d u + d ( s + 1 , r ) The “p arent” no de is deleted from the stack and the children are inserted in the stack and sorted with respect to distanc e, or discarded if the distance is g reater than T . (Remark that since you know the minimum dis- tances component-wise, you can r equire a stronger condition without losing op timality , namely , d ( u,r ) + P n t = s +1 d ( t, j 1 ( t )) < T ). • If s = k , gen erate th e Reed-Solomo n codeword v = ( v 1 , . . . , v n ) = G u and stor e ( n, v , d v ) in th e stack (recall that u is an in itial segment of v ), where d v = d u + n X t = k +1 d ( t, v t ) • If s = n , the sear ch term inates and the initial segment of length k of u is th e decoded message. d) Choice of the cost functio n T : A simple boun d fo r the decoder may be the distance from the received signal of the (uniqu e) Golden- RS cod ew o rd corre sponding to the “closest choice”  U ( j 0 (1)) , . . . , U ( j 0 ( k ))  for the first k compo nents. Any subset of k co mpon ents may be used as w ell to improve the minimum pr ovided th at the corresp onding lines in the Reed-Solomo n gen erator matrix are linearly independent. C. S imulation r esults In the 4 - QAM c ase, th e spectral ef ficien cy of th e Golden Reed-Solomo n cod es is gi ven by 8 k b its 2 n channe l u ses = 4 k n bpcu 6 9 12 15 10 −5 10 −4 10 −3 10 −2 10 −1 SNR FER Golden−RS(4,2,3) with ML decoding Golden−RS (4,2,3) with suboptimal decoding Uncoded Golden Code constant for 4 blocks at 2bpcu Fig. 3. Comparison between suboptimal decoding and ML decoding for the RS (4 , 2 , 3) code at 2 bpcu. The first method achie ves a gain of only 1 . 1 dB ov er the uncoded case, compared to the 6 . 1 dB of the second. From Lem ma 2, we get a lower bound for ∆ min : using an ( n, k , d min ) Reed-Solomon code, we ha ve ∆ min ≥ δ d 2 min . If k = n 2 , th e sp ectral efficiency is 2 bpcu. Com paring th e 4 - QAM, ( n, k , d min ) Golden-RS design ( E S = 0 . 5 ) with the uncod ed Golden Cod e using BPSK ( E S ,U = 0 . 25 ), we g et an asymptotic coding gain of: γ as = √ ∆ min /E S p ∆ min ,U /E S ,U = d min / 0 . 5 1 / 0 . 25 = d min 2 (24) Figures 3 an d 4 show the per forman ce c omparison s of the Golden-RS co des (4 , 2 , 3) an d (6 , 3 , 4) with the corr espondin g uncod ed schem es at the spectral efficiency of 2 b pcu. Assuming the channel to be constant for 4 blo cks and 6 block s respectively , th e Golden-RS codes o utperf orm the un coded scheme by 6 . 1 dB and 7 . 0 dB . The gain for the (4 , 2 , 3) code is unexpe ctedly high com pared with the theo retical co ding gain (24) fo r d = 3 , that is 10 log 10  3 2  dB = 1 . 7 dB . The rough estimate (24) is based on the worst possible o ccurren ce, that o f a cod ew o rd of Hamming weight 3 in which all three non-zer o components correspon d to in vertible elements in the q uotient. Howe ver, we ca n verify em pirically that in the 4 -QAM case and with o ur choice of the (4 , 2 , 3) code, this event does not take place and in fact the ac tual value f or ∆ min found by computer search is 18 , giving an estimate for the gain of 3 . 2 dB , a little c loser to the observed value. This fa vorable b ehavior m ight be du e to the fact that th e chosen co nstellation contains o nly one point in each coset, so that the codew o rds of Ha mming distance 3 are fe w . Also for the (6 , 3 , 4 ) co de, the actual gain ( 7 . 0 dB ) is hig her than the theor etical gain ( 10 log 10 2 dB = 3 . 0 dB based solely on the minim um Hamming distance; 5 . 3 dB using the tr ue value o f ∆ min , that is 46 .) D. Sub -optima l decod ing One can r eplace ML d ecoding with n separate Sphere Decoders on each of the n c ompon ents of Y . The s ig nal is then demodu lated, and mapp ed to a vector ( ˆ V 1 , . . . , ˆ V n ) in F n 256 8 Submitted to IEEE T ra ns. on Inform. The ory , Dec. 2007 6 8 10 12 14 16 10 −7 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 SNR FER Uncoded Golden Code constant for 6 blocks at 2bpcu Golden−RS (6,3,4) with hard decoding Golden−RS(6,3,4) with soft decoding Fig. 4. Comparison between suboptimal decoding and ML decoding for the RS (6 , 3 , 4) code at 2 bpcu. The first method achie ves a gain of 2 . 4 dB over the uncoded case, compared to the 7 . 0 dB of ML decoding. 6 9 12 15 18 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 SNR FER Golden−RS(4,2,3) Uncoded GC, constant for 4 blocks Golden−RS(8,4,5) Uncoded GC, constant for 8 blocks Golden−RS(12,6,7) Uncoded GC constant for 12 blocks Fig. 5. Performanc e of (4 , 2 , 3) , (8 , 4 , 5) , and (12 , 6 , 7) Golden Reed- Solomon codes with suboptimal decoding at 2 bpcu compared with the uncoded Golden Code scheme with the same spectral efficie ncy . using the inverse map pings o f Steps 3 an d 2 in Sectio n VI-A. The recei ved seque nce ( ˆ V 1 , . . . , ˆ V n ) do esn’t n ecessarily belong to the RS code, so a final st ep of RS decoding is needed. This “ hard” decoding has the ad vantage of speed and allows to use longer Reed-Solomo n cod es with high minimu m distance. Howe ver it is highly subop timal; perfor mance simu- lations show tha t with this meth od the c oding g ain is almost entirely cancelled out (see figure 3). Suboptima l dec oding also pr ovides a goo d in itial bo und of the distance of the received poin t from the lattice, which can be used as a cost fun ction for the stack d ecoder described in Section VI-B.1. • 2 bpcu: Figure 5 shows the perform ance co mparison of the Golden-RS codes with suboptimal dec oding with the uncod ed schem e at the spectral ef ficiency of 2 bpcu. Assuming the chann el to be co nstant f or 4 , 8 an d 1 2 blocks re spectiv ely , th e (4 , 2 , 3) , (8 , 4 , 5 ) and (1 2 , 6 , 7) Golden-RS c odes outperform the uncoded scheme at the same sp ectral ef ficiency by 1 . 1 dB , 1 . 7 dB and 2 . 8 dB at the FER of 10 − 3 . 6 9 12 15 18 10 −4 10 −3 10 −2 10 −1 10 0 SNR FER Golden−RS(8,6,3) Uncoded GC constant for 8 blocks Golden−RS(16,12,5) Uncoded GC constant for 12 blocks Golden−RS(24,18,7) Uncoded GC constant for 24 blocks Fig. 6. Perfor mance of (8 , 6 , 3) , (16 , 12 , 5) , and (24 , 18 , 7) Golden Reed- Solomon codes with suboptimal decoding at 3 bpcu compared with the uncoded Golden Code scheme with the same spectral efficie ncy . The Golden-RS schem es seem to be mor e robust on slow fading chann els; in fact the perform ances o f the Golden- RS ( n, k , d min ) codes on a channel which is constant for n block s remain alm ost unch anged (the variation is less than 0 . 2 dB ) when n v aries b etween 4 and 12 , while the uncod ed Gold en Code has a loss o f almost 1 . 5 dB . • 3 bpcu: Assuming the chan nel to b e co nstant for 8 , 1 6 and 2 4 blocks respectively , th e (8 , 6 , 3 ) , (1 6 , 1 2 , 5 ) and (24 , 1 8 , 7 ) Golden- RS codes gain 1 . 5 dB , 2 . 2 dB and 2 . 8 dB over the unco ded sch eme at the FER of 10 − 3 (see Figure 6). Similarly to the previous case, the Golden- RS ( n, k , d min ) codes lose less than 0 . 3 dB wh en n varies between 8 and 24 , while the Golden Code h as a loss of 1 . 1 dB . E. The 16 -QA M case Using 16 -QAM modu lation for each symb ol a, b, c , d in a Golden codeword, there are 2 16 av ailable Gold en code words, or 256 words for each o f the 256 cosets of 2 G in G . As in the 4 -QAM case, we con sider co set co des where the outer code is a n ( n, k , d min ) Reed-Solo mon c ode C on the quotient G / 2 G . Intuitively , the min imum d istance of the Reed-Solomo n c ode “pr otects” the cosets from b eing deco ded wrongly ; if this ch oice is cor rect, the estimate f or the righ t point in the co set is pro tected b y th e minim um determinan t in 2 G . The total infor mation bits transmitted ar e 8 k + 8 n ; they will be encoded into 8 n + 8 n = 1 6 n bits. - The cod e C outputs 8 n bits, which are used to enco de the first two bits of 4 n 1 6 -QAM co nstellations, that is the bits which identify o ne of the four cosets of 2 Z [ i ] in Z [ i ] ; each byte cor respond s to a different coset configur ation of ( a, b, c, d ) (see Figure 8). - the other 8 n b its, left uncod ed, are used to ch oose the last two bits of each 16 - QAM signal. In total, we have 4 n 16 - QAM symbols, that is a vector of n Golden codewords X = ( X 1 , . . . , X n ) . Th e resulting spectral 9 Submitted to IEEE T ra ns. on Inform. The ory , Dec. 2007 k bytes   RS( n, k , d )   n b y tes  % % J J J J J J J J J J J J n bytes “uncoded”  y y t t t t t t t t t 2 n b ytes  ⌣ ⌣ ⌣ ⌣ 4 n 16-QAM symbols = n Golden codewords Fig. 7. The output of the Reed-Solomon code and the uncoded bits are “mingled ” before modulation . 1100 1000 0100 0000 1101 1001 0101 0001 0110 1010 1110 0010 1111 1011 0111 0011 Fig. 8. The la belling of the 16 -QAM constel lation used for performance simulatio ns. The first and second bit identify one of the four cosets of 2 Z [ i ] in Z [ i ] (drawn in dif ferent sha des of gray); the third a nd fourth bit id entify one of the four points in the coset . W e remark that this type of labelli ng cannot be a Gray mapping. efficiency is 8( k + n ) bits 2 n channel uses = 4( k + n ) n bpcu In this case, the coding gain depen ds on the minim um Ham- ming distance inside each c oset in a ddition to the minimum Hamming distance in the quotient: we ha ve seen in (22) that ∆ min ≥ min  min X ∈ 2 G \{ 0 } , d 2 min  = min(16 , d 2 min ) (25 ) W ith a n erro r-correcting cod e of rate k = n 2 , we o btain a spectral effi ciency o f 6 bpcu. - If d min ≥ 4 , we have γ as = 4 / 2 . 5 1 / 1 . 5 = 2 . 4 , lead ing to an app roximate gain o f 3 . 8 dB . Th us it d oes not seem worthwhile to use lo ng codes with a hig h min imum distance with this scheme. - If d min = 3 , γ as = 3 / 2 . 5 1 / 1 . 5 = 1 . 8 , m aking for a gain o f 2 . 5 dB . Decoding The ML decod ing pro cedure f or the 16 - QAM case requires only a slight modificatio n with respect to Step 6 illustrated in Section VI-B. In the first phase, fo r each co mponen t i = 1 , . . . , n and for e ach coset lea der W j , j = 0 , . . . , 2 55 , we find the closest point in that coset to the recei ved compon ent Y i , that is ˆ X i,j = argmin X ∈ 2 G k Y i − H ( X + W j ) k 2 6 8 10 12 14 16 18 10 −4 10 −3 10 −2 10 −1 10 0 SNR FER Uncoded GC, channel constant for 4 blocks Golden−RS (4,2,3) using 16−QAM Uncoded GC, channel constant for 6 blocks Golden−RS (6,3,4) using 16−QAM Fig. 9. Performance of the (4 , 2 , 3) and (6 , 3 , 4) Golden Reed-Solomon codes with ML decoding at 6 bpcu compared with the uncoded schemes with the s ame spectral effic ienc y . Computing H X and H W j separately allo ws to perform only 512 pro ducts instead of 2 5 6 2 . T he second ph ase can be perfor med as in the 4 -QAM case, an d the search is limited to the “closest poin ts” ˆ X i,j + W j determined in the previous phase: ˆ X = argmin ( ˆ X 1 ,j 1 + W j 1 ,..., ˆ X n,j n + W j n ) n X i =1    H ( ˆ X i,j i + W j i ) − Y i    2 over all the images ( W j 1 , . . . , W j n ) of Reed -Solomo n code- words. Simulation r esults In the 16 -QAM case, the (4 , 2 , 3) and (6 , 3 , 4) Golden Reed-Solomo n co des ach iev e a g ain of 3 . 9 dB and 4 . 3 dB respectively over the uncoded scheme at 6 bpcu at the fram e error rate of 10 − 2 , supposing that the channel is constan t for 4 and 6 time blocks ( see figure 9). V I I . C O N C L U S I O N S In this paper we have presented Golden-RS codes, a cod ed modulatio n scheme for 2 × 2 slow fading MIMO chann els, where the inner code is the Golden Cod e. W e u se a simple b inary partition ing, whose set of co set leaders coincides with a QAM symbol co nstellation. W ith a Reed - Solomon code as the outer co de in order to increase the minimum Hamm ing distance among the cod ew o rds, we obtain a significan t per forman ce gain with respect to the uncod ed case. A P P E N D I X I P R O O F S W e rep ort here some of the pr oofs for the results stated in the main part of the p aper . 10 Submitted to IEEE T ra ns. on Inform. The ory , Dec. 2007 Pr oof of Lemma 1. For all i = 1 , . . . , L , let Q i = X i X H i : then det( X 1 X H 1 + . . . + X L X H L ) 1 = = det( Q 1 + . . . + Q L ) 1 = = ( e Q 1 + . . . + e Q L )( Q 1 + . . . + Q L ) 1 = = L X i,j =1 e Q i Q j = L X i =1 det( Q i ) 1 + X i 6 = j e Q i Q j W e nee d to s how th at e Q i Q j + e Q j Q i =    e X j X i    2 F 1 . But k X k 2 F = tr( X X H ) , and therefore    e X j X i    2 F = tr( e X j X i X H i e X H j ) , and e Q j Q i = e X H j e X j X i X H i , e Q i Q j = ^ e Q j Q i ⇒ e Q i Q j + e Q j Q i = tr( e Q i Q j ) 1 = tr( e X j X i X H i e X H j ) 1 , recalling that tr( AB ) = tr( B A ) . Pr oof of Remark 3. a) Let W =  w 1 w 2 i w 2 w 1  , w 1 = t 1 + is 1 , w 2 = t 2 + is 2 , where t 1 , t 2 , s 1 , s 2 ∈ Z [ θ ] . Then k W k 2 F = | w 1 | 2 + | w 1 | 2 + | w 2 | 2 + ¯ w 2 2 . But w 1 = a + bθ + i ( c + dθ ) for some a, b, c , d ∈ Z , and | w 1 | 2 + | w 1 | 2 = = ( a + bθ ) 2 + ( c + dθ ) 2 + ( a + b ¯ θ ) 2 + ( c + d ¯ θ ) 2 = = 2 a 2 + 3 b 2 + 2 ab + 2 c 2 + 3 d 2 + 2 cd ∈ Z The same is true for | w 2 | 2 + | w 2 | 2 . b) If X =  a b c d  , then k X k 2 F = | a | 2 + | b | 2 + | c | 2 + | d | 2 ≥ 2 ( | ad | + | bc | ) ≥ 2 | ad − bc | = 2 | det( X ) | and    e X Y    2 F ≥ 2    det( e X Y )    = 2 | det( X ) det( Y ) | c) Let X 1 = 1 √ 5 AW 1 , X 2 = 1 √ 5 AW 2 , W 1 , W 2 ∈ O . Then    e X 2 X 1    2 F = 1 25    f W 2 e AAW 1    2 F = | N ( α ) | 2 25    f W 2 W 1    2 F = 1 5    f W 2 W 1    2 F ≥ 2 5 , since W = f W 2 W 1 belongs to O . A P P E N D I X I I Q UAT E R N I O N A L G E B R A S This section summarizes some b asic facts ab out q uaternion algebras that are used in the p aper . Our main references are the books of V ig n ´ eras [12] and Rein er [9]. Definition 1 ( Quate rnion algebra s ) . Let K b e a field. A quaternio n algebra H of center K is a central simp le alg ebra of dimensio n 4 over K , such that th ere exists a separa ble quadra tic extension L of K , an d an element γ ∈ K ∗ , such that H = L ⊕ Le, e 2 = γ , ex = σ ( x ) e ∀ x ∈ L where σ is the non-trivial K -automo rphism of L . L is called a maximal subfield of H . H will be deno ted b y the triple ( L/K , σ, γ ) . Quaternion algebras are a special case of cyclic a lgebras . T o obta in a rep resentation of H as a K - module, c onsider a primitive element i such that L = K ( i ) , and let j = e , k = ij = j σ ( i ) . Then H = { a + bi + cj + dk | a, b, c, d ∈ K } (26) The fo llowing theorem giv es a suf ficient con dition for a quaternio n alge bra to be a d ivision rin g: Theorem 8 . Let H = ( L/K , σ, γ ) be a quaternion algebra. If γ is n ot a reduced norm of any element of L , then H is a skew fi eld. Definition 2 ( Splitting fields ) . Let H be a central simp le K - algebra. An extension field E of K sp lits H , or is a splitting field fo r H , if E ⊗ K H ∼ = M r ( E ) In the case of division algebras, every maximal subfield is a splitting field: Theorem 9. Let D be a ske wfield with center K , with fi nite de g r ee over K . Then every maximal subfie ld E of D contain s K , an d is a splitting field fo r D . In the following parag raphs we will always co nsider a Dedekind doma in R , its quotient field K , and a quaternio n algebra H over K . Definition 3 ( Latt ices and orders ) . A full R -lattice or ideal in H is a finitely genera ted R -submod ule I in H such that K I = H , where K I = ( n X i =1 k i x i    k i ∈ K , x i ∈ I , n ∈ N ) An R -order Θ in H is a fu ll R -lattice wh ich is also a s ubring of H with the same un ity element. A maximal R -order is an order which is not prope rly con tained in any other order o f H . For the fo llowing pro position see for example Reiner [9]: Proposition 10. A sub ring of H co ntaining a basis fo r H over K is an or der if and on ly if all its elements a r e inte g ral over R . 11 Submitted to IEEE T ra ns. on Inform. The ory , Dec. 2007 Remark 6. Th e notion of or der is a gen eralization o f the notion of the ring of integers for comm utativ e extensions. Howe ver, in th e non-c ommutative case the set of elements which are integral over the base field might not b e a ring. Definition 4 ( Propert ies of id eals ) . Gi ven an ideal I of H , we can define the left or der an d the right or der of I as fo llows: Θ l ( I ) = { x ∈ H | I x ⊂ I } , Θ r ( I ) = { x ∈ H | xI ⊂ I } Θ l ( I ) and Θ r ( I ) are orders. I is called • two-sided if Θ l ( I ) = Θ r ( I ) , • normal if Θ l ( I ) and Θ r ( I ) are m aximal, • integr a l if I ⊂ Θ l ( I ) , I ⊂ Θ r ( I ) , • principa l if I = Θ l ( I ) x = x Θ r ( I ) for some x ∈ H The inverse of I is the fractional id eal I − 1 = { x ∈ H | I xI ⊂ I } . The norm N ( I ) of an ideal I is the set of reduced nor ms of its elements, an d it is an ideal of R . If I = Θ x is princip al, N ( I ) = R N ( x ) . A P P E N D I X I I I I D E A L S , V A L U A T I O N S A N D M A X I M A L O R D E R S Definition 5 ( Pr ime ideals ) . Let Θ be an ord er , P a two-sided ideal of Θ (that is, the left a nd right or der o f I coincide with Θ ). P is prime if it is nonzero and ∀ I , J integer two-sided ideals of Θ , I J ⊂ P ⇒ I ⊂ P or J ⊂ P . The p roofs of th e f ollowing theor ems can b e fou nd in Reiner’ s book [9]: Theorem 11. The two-sided ide als of an or der Θ form a fr ee gr oup generated by the p rime ideals. Theorem 1 2. Let Θ be a maximal o r der in a quaternio n algebra H . Then the prime ideals of Θ coincide with the maximal two-sided ideals of Θ , and ther e is a one- to-one corr espo ndence between the p rime ideals P in H and the prime ideals P of R , given by P = R ∩ P . Mor eover , Θ / P is a simple algebra over the fin ite field R /P . Definition 6 ( V aluations and local fields ) . A valuation v of K is a positi ve real function of K su ch that ∀ k, h ∈ K , 1) v ( k ) = 0 ⇔ k = 0 , 2) v ( k h ) = v ( k ) v ( h ) , 3) v ( k + h ) ≤ v ( k ) + v ( h ) . v is non-arc himedean if v ( k + h ) ≤ max( v ( k ) , v ( h )) ∀ k , h ∈ K ; it is discr ete if v ( K ∗ ) is an infinite cyclic g roup. K can be endowed with a top ology indu ced by v in the following way: a neighb orhoo d basis of a point k is given by the sets U ε ( k ) = { h ∈ K | v ( h − k ) < ε } K will be called comp lete if it is co mplete with resp ect to this topolog y . If v is non archimedean, the s et R v = { k ∈ K | v ( k ) ≤ 1 } is a local rin g, called the valu ation ring of v . The quotien t R v /P v , where P v is the un ique maximal ideal of R v , is called the field of r esidues of K . K is a local field if it is complete with r espect to a discrete valuation v and if R v /P v is finite. Definition 7 ( Places ) . A place v of K is an immersion i v : K → K v into a local field K v . I f v is non -archime dean, we say that it is a finite plac e ; otherwise, th at it is an infinite place . The finite places of K arise fro m d iscrete P -a dic valuations of K , where P ran ges over the max imal ideals in the ring of integers R of K . (Recall th at the ring o f integers in a n umber field is always a Ded ekind d omain, an d so th e maximal ideals coincide with the prime ideals). Definition 8 ( Ramified places ) . Let H be a quaternion algeb ra over K , and P a place of K . Consider th e K -mo dule H P = H ⊗ K K P ; H P is is omorp hic to a matrix alg ebra M r ( D ) over a skew field D of cen ter K P and index m P over K P ; m P is called the lo cal index of H a t P . W e say tha t P is ramified in H if m P > 1 . Giv en a max imal order Θ , the set Ram( H ) of ram ified places of H is related to a particular tw o-sided ideal of Θ : Definition 9 ( Different a nd discriminant ) . Let Θ be an ord er . The set Θ ∗ = { x ∈ H | tr ( x Θ) ⊂ R } is a two-sided idea l, called the d ual of Θ . Its inv e rse D = (Θ ∗ ) − 1 is a two-sided integral ideal, called the differ ent of Θ . If { w 1 , . . . , w 4 } is a basis of Θ as a f ree R -mo dule, ( n ( D )) 2 = R det (tr( w i w j )) The ideal n ( D ) of R is called the r edu ced d iscriminant of Θ and is denoted by d (Θ) . Proposition 13. If Θ , Θ ′ ar e two or ders and Θ ′ ( Θ , then d (Θ ′ ) ( d (Θ) . The n otion of ramification for quater nion algebr as is a gen- eralization of the notion of r amification for field extensions: Theorem 14. Let Θ b e a maximal o r der in H . F or each place P of K , let m P be the local index of H at P , an d let P be the prime ideal of Θ co rr espon ding to P (see The or em 12). Then m P > 1 only for a finite number of p laces P , and P Θ = P m P , D = Y P ∈ Ram( H ) P m P − 1 Proposition 15 . Let H b e a quate rnion algebra unramified at infinity . A necessary an d su fficient con dition for an order Θ to be maximal is that d (Θ) = Y P ∈ Ram( H ) \∞ P In the case of infinite places P , the P -adic co mpletion can be R ( r eal primes ) or C ( complex primes ). Comp lex primes are nev er ramified [9]. 12 Submitted to IEEE T ra ns. on Inform. The ory , Dec. 2007 Theorem 16. The two-sided ideals of a max imal or der Θ form a commutative gr ou p with r espect to multiplica tion, which is generated by the ideals of R an d the ideals of r ed uced norm P , whe r e P varies over the prime id eals of R that are r amified in H . R E F E R E N C E S [1] J-C. Belfiore, G. Rekaya, E. Vi terbo, “The Golden Code: a 2 × 2 full- rate Space-T ime Code with non-v anishing determin ants”, IEEE T rans. Inform. Theory , vol 51 n.4, 2005 [2] S. Benedett o, E. 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Reiner , “Maximal Orders”, Clarendon P ress, Oxford 2003 [10] B. A. Sethurama n, B. S. Rajan, V . Shashidar , “Full-di versity , hi gh-rate space-t ime bloc k codes from di vision alge bras”, IE EE T rans. Inform. Theory , vol 49, 2003 [11] V . T arokh, N. Seshadri, A. R. Calderbank, “Space -time codes for high data rate wireless communicat ion: performance criterion and code construct ion”, IEEE T rans. Inform. Theory , vol. 44 no. 2, 1998 [12] M-F . V ign ´ eras, “ Arithm ´ etique des Alg ` ebres de Quaterni ons”, Lecture Notes in Mathemati cs, Sprin ger V erl ag 1980 13