Space-Time Codes from Structured Lattices

SUBMITTED TO IEEE TRANS. INFORM. THEOR Y , APR. 2008 1 Space-T ime Codes from Structured Lattice s K. Raj Kumar and Giuseppe Caire Abstract W e present constructions of Sp ace-Time (ST) codes b ased on lattice co set codin g. First, we focu s on ST code constructions fo r the short block-length case, i.e., wh en the block- length is equal to or slightly larger than the nu mber of transmit antenn as. W e p resent construction s b ased on den se lattice packing s and ne sted lattice (V oronoi) shaping. Our codes achieve the op timal div ersity-multip lexing tradeoff of quasi-static MIMO fading chan nels for any fadin g statistics, a nd perf orm very well a lso at practical, moderate values of signal to n oise ratios (SNR). Then, we extend the constructio n to the case of large bloc k lengths, by using trellis coset coding . W e provid e constructio ns of trellis cod ed mod ulation (TCM) schemes tha t are endowed with good p acking and sh aping prop erties. Both short- block an d trellis construction s allow for a reduced complexity decoding algorithm based on minimum mean squared error g eneralized decision feedb ack equalizer ( MMSE-GDFE) lattice deco ding and a combin ation of this with a V iterbi TCM decoder for the TCM c ase. Beyond the interesting algeb raic structure, we exhibit codes whose perfor mance is amon g the state-o f-the art co nsidering cod es with similar enco ding/deco ding complexity . The authors are wi th the Department of Electrical Enginee ring - Systems, Univ ersity of Southern California, Los Ange les, CA 90089, USA ( { rkkrishn,ca ire } @usc.edu ). The material in this paper was presented in part at the IEEE International Symposia on Information T heory ISIT -2006 in Seattle, USA and ISIT -2007 i n Nice, France. This work has been partially f unded by a gift from ST Microelectronics, NSF Grant No. CCF-0635326, the 2006 Okaw a Founda tion Research Grant and by an Oakley fellowship from the G raduate School at USC. October 29, 2018 DRAFT SUBMITTED TO IEEE TRANS. INFORM. THEOR Y , APR. 2008 2 I . I N T R O D U C T I O N The qu asi-static, frequency-flat fading (complex) multiple-input multiple-output (MIM O) c hannel with M trans mit and N receiv e antenna s and coding bloc k-length T chan nel uses is de scribed by Y c = H c X c + W c , (1) where X c denotes the M × T transmitted codewor d ma trix drawn from a s pace-time (ST) c ode X , Y c is the N × T r eceived sign al matrix, H c is the N × M chann el ma trix and W c is the N × T noise matrix. The entries of the channel matrix H c are assumed to be constan t over a b lock length of T channel uses and the entries of W c are indep endent and ide ntically distributed complex Gauss ian with zero mean and unit variance, i.e., i.i.d. CN (0 , 1) . Th e resu lts o f this paper will hold for arbitrary cha nnel fading statistics, but we w ill u se the standard i.i.d. Ra yleigh fading model for our s imulations, in which case the entries of H c are i.i.d. CN (0 , 1) . The input c onstraint E k X c k 2 F ≤ T SNR (2) is enforced, where E ( · ) de notes the expectation op erator and SNR takes on the meaning of the transmit signal-to-noise ratio (total transmit energy per cha nnel use over the no ise power spectral dens ity). The channe l matrix H c is as sumed to be kn own perfectly at the re ceiv er but not at the transmitter . The use of S T code s over MIMO channels is kno wn to provide two kinds of be nefits: better reliability through div ersity gain, and h igher d ata rates in terms of multi plexing gain. The diversity-multi plexing tradeoff (DMT) (see [9] for the de finition a nd details) captures in a succinct and e legant way the tradeo f f between these two quantities in the high signal to noise ratio (SNR) regime. T he DMT specifie s the maximum possible div ersity that can be obtained at ea ch po ssible value of multiplexing gain, a nd ha s become a s tandard performance metr ic to e valuate ST schemes, and a tool to compare different ST scheme s. Families of codes that ac hiev e the DMT of MIMO fading cha nnels have bee n propose d. Perhap s the most notable in terms of pe rformance a nd ge nerality a re La ttice ST (La ST) c odes and c odes ob tained from cyclic division algebras (CD A). An ensemb le of randomly ge nerated LaST codes was s hown to be DMT optimal un der minimum me an squared error generalized decision f eedba ck equalizer (MMSE-GDFE) latt ice dec oding for T ≥ M + N − 1 [1]. In this c ase, DMT o ptimality is shown in a rand om coding s ense (i.e., with res pect to e rror probability av eraged over the random lattice ensemble) and for the Rayleigh i.i.d. fading statistics. Families of carefully c onstructed CD A code s en joy the so-ca lled n on-vanishing determina nt (NVD) property (to be defined subsequently), which in turns implies that these codes, under ML decod ing, achieve October 29, 2018 DRAFT SUBMITTED TO IEEE TRANS. INFORM. THEOR Y , AP R. 2008 3 the optimal DMT in a un i versal sense, i.e., over any chann el f ading statistics [2]. Codes ach ieving the optimal DMT over any fading statistics are c alled “approximately uni versal” in [3]. F urthermore, these codes allow for minimum b lock length, i.e. , there exist op timal codes for all T ≥ M [2]. In some sense, the prese nt work may be thou ght of as a confluenc e of t hese two a pproache s. W e construct codes that retain desirable properties from b oth famil ies: n ot only are they are non-rand om explicit constructions from CD As, but they also employ the nested lattice construction that enables shaping gains and the reduced complexity MMSE-GDFE lattice decoding akin to the LaST c odes. The DMT ca ptures the optimal p erformance for high SNR. Follo wing [1], [2], attention has shifted tow ards constructing S T codes tha t not on ly achiev e the DMT , but also perform well at finite (practical) values of SNR. For example, generating c odes at rando m f rom the ensemble of [1] yields typically performances that stay at 1 to 3 d B from outage probability (that can b e regarded an effecti ve “quasi- lower boun d” on the performance of any code a t meaningful SNR, i.e ., for probability o f block error not too lar ge (say , ≤ 10 − 1 )). In this perspe cti ve, the first pa rt our this work prese nts a construction o f structured LaST (S-LaST) c odes 1 that achieve the DMT and p erform well at finite S NR, for small to moderate b lock-lengths (i.e., T is e qual to or slightly larger than M ). In the secon d part of the pa per we turn to the case of large block len gths T ≫ M . This is motiv ated by the fact that i n practical wireless c ommunication sy stems, information is enc oded and s ent over the c hannel in packets, togethe r with training s ymbols, protocol information, and guard intervals. Therefore, pa ckets c annot be too small, for othe rwise the overhead would be a lar ge part of the overall cap acity . W e target the c ase whe re da ta packets span a number of channel uses T considerab ly larger tha n the n umber of transmit antenna s M , but nevertheless sma ller than a fading coheren ce interv al. Then, the fading ch annel is constan t over the whole codeword of duration T c hannel uses . Unfortunately , the LaST and/or CD A constructions d o not gene ralize, in p ractice, to T ≫ M since the decoding complexity grows rapidly with T . Furthermore, with constructions s uch a s those in [1], [2] it is not clear h ow to exploit the large block len gth to o btain codes with improved co ding g ain. Therefore, the c hallenge he re is to design ST codes for lar ge T that have good co ding gain and low decoding complexity . In this regard, the autho rs in [21] have proposed a trelli s code d modulation (TCM) scheme based on partitions o f the Golden c ode [11 ]. For prior work on ST T CM, s ee [18], [19]. Bu ilding on these ideas, we propos e a general technique for the construction of ST -TCM s chemes with good coding and sha ping ga ins. Thes e co des can be de coded using the V iterbi Algorithm where the branch metrics are 1 W e use the term “structured” to distinguish these codes from the random l attice approach of [1]. October 29, 2018 DRAFT SUBMITTED TO IEEE TRANS. INFORM. THEOR Y , AP R. 2008 4 computed using a low complexity MMSE-GDFE lattice decoder . W e sho w construction examples based on the Gosset lattice E 8 and lattices drawn from the Golde n+ algeb ra [12] that yield, to the best of the authors’ k nowledge, t he current s tate-of-the art performance a mong cod es with s imilar enc oding/deco ding complexity . In Se ction II we revie w LaS T cod es and ST codes from CD As, as these form the two main ing redients for our con struction. W e a lso revie w some con cepts relating to lattice p ackings tha t will be use d su b- sequen tly . Code design for the short bloc k-length case is pres ented in Section III, and Sec tion IV d eals with the c onstruction o f TCM sc hemes. Simulations resu lts are provided along side ea ch construction, and illustrate the eff ectiveness of the constructions. I I . B AC K G R O U N D A. La ttice Spac e-T ime (LaST) cod es An n -dimensiona l real lattice Λ is a discrete additi ve s ubgroup of R n defined as Λ = { Gu : u ∈ Z n } , where G is the n × n (full-rank) rea l generator matrix of Λ . T he fundamen tal V oronoi cell of Λ , denoted as V (Λ) , is the s et of points x ∈ R n closer to zero than to any other p oint λ ∈ Λ . The fundamen tal volume of Λ is V f (Λ) , V ( V (Λ)) = Z V (Λ) d x = q det ( G T G ) . An n -dimensiona l lattice cod e C (Λ , u 0 , R ) is the finite su bset of the lattice translate Λ + u 0 inside the shaping region R , i.e., C = { Λ + u 0 } ∩ R , where R is a b ounded meas urable region of R n . LaST codes a re more easily illustrated by conside ring the real vectorized channel mode l equ i valent to (1), y = H x + w , (3) where x ∈ R 2 M T and y , w ∈ R 2 N T denote respecti vely the vector equiv a lents of X c , Y c and W c obtained by separating real and imaginary part and by stacking columns, and where H = I T ⊗   Re( H c ) − Im( H c ) Im( H c ) Re( H c )   , according to the well-kno wn construction as in [1] . W e say t hat an M × T sp ace-time coding scheme X is a full-dimensional LaST c ode if it’ s vectorized (rea l) code book (corres ponding to the channe l model in (3)) is a lattice code C (Λ , u 0 , R ) , for some n -dimension al lattice Λ , translation vector u 0 , and sh aping region R , where n = 2 M T . G i ven the equi valence of the real vector an d the c omplex ma trix rep resentation of X , we shall not distinguish b etween them explicitly and write simply X = C (Λ , u 0 , R ) . Any linear- dispersion ST code, including the con structions of [2], can be represen ted as a LaST code , for a suitable October 29, 2018 DRAFT SUBMITTED TO IEEE TRANS. INFORM. THEOR Y , AP R. 2008 5 shaping region. For later use, we define the lattice quantization func tion as Q Λ ( y ) , arg min λ ∈ Λ | y − λ | and the mo dulo-lattice function [ y ] mod Λ = y − Q Λ ( y ) . W e also define the n otion of a non-vanishing determinan t (NVD) for an infi nite L aST c ode (i.e., disre- garding the sh aping region R ) a s follows. A LaST code has the NVD p roperty if and only if the minimum determinant correspond ing to its infin ite latti ce Λ is bounded away from zero by a c onstant independen t of SNR, i.e. , 2 min ∆ X c = X c i − X c j , x i 6 = x j , x i , x j ∈ Λ + u 0 det h ∆ X c (∆ X c ) H i ˙ ≥ SNR 0 . Notice that s ince Λ is a lattice, this is equiv alent to min x ∈ Λ+ u 0 det h X c ( X c ) H i ˙ ≥ SNR 0 . B. ST Codes fr om CD A For a detailed e xposition of ST codes from CD A, we r efer the read er to [24], [2] and references therein. W e provi de a very b rief review in the seque l. Let Q den ote the field of rational n umbers a nd ı , √ − 1 . Se t F = Q ( ı ) . The cons truction o f a CD A c alls for the c onstruction of an n -degree cyc lic Galois extension L / F with ge nerator σ . Then a CD A D ( L / F , σ, γ ) with ce nter F , maximal subfi eld L and index n is the set of all elements of the form P n − 1 i =0 z i ℓ i , wh ere z is an indeterminate sa tisfying ℓz = z σ ( ℓ ) ∀ ℓ ∈ L an d z n = γ . The eleme nt γ need s to be a prope rly ch osen non-norm elemen t in order to ensure that D is a division algebra, s ee [24], [2] for details. Every e lement in the CD A can be assoc iated with an n × n ma trix through the left re gular repr e sentation , which is of the form         ℓ 0 γ σ ( ℓ n − 1 ) γ σ 2 ( ℓ n − 2 ) . . . γ σ n − 1 ( ℓ 1 ) ℓ 1 σ ( ℓ 0 ) γ σ 2 ( ℓ n − 1 ) . . . γ σ n − 1 ( ℓ 2 ) . . . . . . . . . . . . . . . ℓ n − 1 σ ( ℓ n − 2 ) σ 2 ( ℓ n − 3 ) . . . σ n − 1 ( ℓ 0 )         , (4) 2 W e make use of the exponential equality notation from [ 9], defined as a . = ρ − b ⇔ b = − lim ρ →∞ log a log ρ . The notations ˙ ≥ and ˙ ≤ are defined similarly . October 29, 2018 DRAFT SUBMITTED TO IEEE TRANS. INFORM. THEOR Y , AP R. 2008 6 where ℓ i ∈ L . The trace and determinant of the above matrix are resp ectiv ely defined to be the r e duced trace tr r ( · ) and reduced norm N r ( · ) of the element it repres ents. The ST c ode with M = T = n is a finite c ollection of matrices of the above form, sc aled to sa tisfy the power constraint in (2 ). Cho osing γ ∈ Z [ ı ] and restricting the ℓ i to belong to the ring o f integers O L of L b estows the NVD property on the ST code. One such choice for the ℓ i correspond s to choosing ℓ i = n X k =1 e i,k β k , e i,k ∈ A QAM , (5) with A QAM = { a + ıb | − Q + 1 ≤ a, b ≤ Q − 1 , a, b odd } , and where β k , k = 1 , 2 , . . . , n is an integral basis (i.e., a basis as a module) for O L / O F . More ge nerally , we could ch oose { β k } n k =1 to c onstitute an O F -basis for a ny ideal I ⊆ O L . In this ca se, | X | = Q 2 n 2 . The resu lts of [2], [3] sh ow that cod es deri ved from CD A with NVD are app r ox imately universal . In the recent work [12], S T codes are obtained from maximal orders in CDAs. For the sake of later use, a brief review follows. A Z [ ı ] − or der in an F − algebra D is a subring O o f D , ha ving the same identity e lement as D , and s uch that O is a finitely gene rated module over Z [ ı ] and gene rates D as a linear spa ce over F . An order O is called maximal if it is not properly con tained in any other Z [ ı ] − orde r . The disc riminant of a Z [ ı ] − order O is comp uted as d ( O /R ) = det ([ tr r ( b i b j )] m i,j =1 ) , where { b 1 , . . . , b m } is any Z [ ı ] − basis of O . All maximal orders of a CD A share the s ame value of the discriminant, an d also have the smallest possible discriminant among all orders within a gi ven CD A. An important property of elements of an order of a CD A D ( L / F , σ, γ ) is that their reduced norm (i.e., the d eterminant of their matrix rep resentation) is an eleme nt of the ring of inte gers O F = Z [ ı ] of the center F . This prope rty ens ures that S T c odes carved out of o rders in suitably con structed CD As are e ndowed with the NVD property . The ch oice of a subset of elements of D correspon ding to (5) amoun ts to choo sing a particular order O kn own as the natural or der . It is e stablished in [12] that the discriminant of an order in a CD A is dir ectly propo rtional to the fundamental volume of the ensuing lattice (they a re in f act eq ual for the c ase whe n the c enter of the CD A is F = Q ( ı ) ). Therefore, in order to maximize the energy efficiency o f the code, a sens ible design guideline is to use the maximal order of the CD A to deriv e S T codes, owing to the m h aving the minimum possible discriminant. All pre vious constructions of ST cod es from CD As, including the on es in [24], [2], [4], [11], [5] have used the natural order , which is not gu aranteed to b e maximal in general. As an illustration of the tec hnique, the authors in [12] c onstruct a 2 × 2 ST code derived from the October 29, 2018 DRAFT SUBMITTED TO IEEE TRANS. INFORM. THEOR Y , AP R. 2008 7 maximal order of a CD A named the Golden + Algebra ( GA +) , whose minimum determinant improves upon that of previously known constructions. W e will revisit this con struction subsequ ently in Section. III, and use it to cons truct some of our examples. C. Lattice P a ckings The classical sphe re pac king problem is to find ho w dens ely a lar ge number of identical spheres can be packed together in n -dimensional spac e. A packing is c alled a lattice pa cking if it h as the property that the se t of centres o f the sph eres forms a lattice in n -dimensional space. An excellent referenc e for this area is the book by Conway and Sloa ne [6]. The den sity ∆ of a lattice pac king is given by ∆ , Proporti on of space that is occu pied by the sph eres = volume of one sp here V f (Λ) . A related q uantity is the center dens ity δ , gi ven by δ = ∆ V n , where V n is the volume o f an n -dimen sional sphere o f radius 1 , gi ven by V n = π n/ 2 ( n/ 2)! = 2 n π ( n − 1) / 2 (( n − 1) / 2 )! n ! (the s econd form avoids the us e of ( n/ 2)! whe n n is odd). A related pa rameter is the fun damental coding gain γ c (Λ) , de fined as: γ c (Λ) , 4 δ 2 /n = d 2 min (Λ) V (Λ) 2 /n , (6) where d min (Λ) denotes the minimum distance of the lattice Λ . It is evident from the defi nition that the fundamental co ding gain is a normalized measu re of the de nsity of the lattice. Further , the fun damental coding ga in also posses ses the des irable prope rties of being dimension less, an d in variant to sca ling and any orthogonal transformation (rotati on) [8]. For the cub ic lattice, γ c ( Z n ) = 1 . The problem of finding de nse p ackings (i.e., tho se with high values of γ c (Λ) ) in n -dimen sional space has a long and interesting history . In two d imensions, Gauss proved tha t the hexagon al lattice is the denses t plan e lattice packing , a nd in 1940 , L. Fe jes T ´ oth proved that the hexagonal la ttice is indee d the denses t of a ll possible p lane p ackings. In 161 1, the German astronomer J ohanne s K epler s tated that no packing in three dimensions can be de nser than that o f the face-centered cubic (f.c.c.) lattice arrangeme nt which fills abou t 0 . 7405 of the av ailable space . It took mathematicians some 400 years to prove him October 29, 2018 DRAFT SUBMITTED TO IEEE TRANS. INFORM. THEOR Y , AP R. 2008 8 right, with T homas Hales proving the conjec ture in 1998 (Gauss showed in 1821 that the f.c. c. lattice is the den sest pos sible lattice pac king in three dimen sions). T he dens est po ssible la ttice pa ckings are known for all d imensions n ≤ 8 . The checkerboard lattices D 4 and D 5 are the de nsest p ossible lattice pac kings in 4 and 5 -dimensions respectively while Gosse t’ s root lattices E 6 , E 7 and E 8 are optimal amon g lattice packings in 6 , 7 a nd 8 -dimensions. It is also known that the de nsest lattice pa ckings in dimen sions 1 to 8 are unique. Although no t proven, it seems likely tha t Coxeter-T odd lattice K 12 , the Barnes-W a ll lattice Λ 16 ∼ = B W 16 and the Leech latti ce Λ 24 are the dense st lattices in dimensions 12 , 16 and 24 respectiv ely [6]. T a bles o f the b est known lattice packings in n -dimensions are available in the literature [6] and in the on line catalogue of latti ces [7]. For later use, we defin e a lattice Λ with gen erator matrix G to be an inte gral lattice if the Gram matrix A , G T G has integer entries. It turns out that many of the b est known lattices in terms of pac king belong to this class, whe n suitably sc aled. I I I . T H E S T RU C T U R E D L A S T C O D E C O N S T RU C T I O N This section dea ls with code des ign for the case of short bloc k-lengths, i.e., T is equa l to or slightly lar ger than M . Be fore we present the cons truction, we first explore the LaST formulation of space-time codes derived from CD A. A. CDA ST Codes as Lattice Codes W e will illustrate the equ i valent lattice structure with an example of a 2 × 2 ST co de de ri ved from CD A. From (4), any c odeword matrix is of the form X c =   ℓ 0 γ σ ( ℓ 1 ) ℓ 1 σ ( ℓ 0 )   . The real vector corresponding to X c in the eq uiv a lent channel model of (3) is given by x = h Re ( x c ) T Im ( x c ) T i T , where x c = [ ℓ 0 ℓ 1 γ σ ( ℓ 1 ) σ ( ℓ 0 )] T ∈ C 4 . October 29, 2018 DRAFT SUBMITTED TO IEEE TRANS. INFORM. THEOR Y , AP R. 2008 9 Let { β 1 , β 2 } denote an integral basis ov er Z [ i ] for s ome ideal I ⊆ O L . Th en, in acco rdance with (5), X c represents a point in the (complex) latti ce whose generator matrix is gi ven by G c =         β 1 β 2 0 0 0 0 β 1 β 2 0 0 γ σ ( β 1 ) γ σ ( β 2 ) σ ( β 1 ) σ ( β 2 ) 0 0         , (7) i.e., x c = G c [ a 1 a 2 a 3 a 4 ] T , { a i } 4 i =1 ∈ Z ( ı ) . The corresp onding real lattice ge nerator matrix is gi ven by G =   Re ( G c ) − Im ( G c ) Im ( G c ) Re ( G c )   . It is now evident that the c hoice o f pa rameters γ and { β 1 , β 2 } completely determines the lattice structure of the ST c ode (as suming a particular generator σ for the g roup of automorphis ms). Furthermore, the choice of these p arameters in conjunction with (5) amoun ts to the choice of a particular subset L of O L to be the signa ling alph abet. The key to ensuring good c onstellation shaping lies in an intelli gent ch oice of the non-norm element and the integral basis. In [4], these parameters are chosen to ens ure that the resultant lattice gene rated by G is a rotated version of the cubic lattice Z 2 M T , i.e., that G is a unitary matrix. The cubic shaping is in fact the best possible shap ing that we ca n obtain by a linear encod er ov er the reals (linear -dispersion c ode). No sh aping gain can be achiev ed by a linear map: a t most, the encoder do es not increa se the transmit en ergy . Th is is indeed obtained by G unitary , that is a n isometry of R 2 M T . The a uthors in [4] provide such con structions for 2 × 2 , 3 × 3 , 4 × 4 an d 6 × 6 (square) ST codes with N VD and have termed the resultant ST codes as perfect codes . More recently , [5] prese nted perfect ST co de c onstructions for a rbitrary number of transmit antennas an d also for the rectangular cas e ( T ≥ M ). B. Th e S-LaST Con struction W e wish to ob tain LaST co des with the follo wing prope rties: 1) t he NVD property; 2) t he un derlying lattice Λ c (referred to a s the cod ing lattice in the following) has large fundamental coding ga in γ c (Λ c ) (se e (6)); 3) t he shaping region R is as close as p ossible to a sphere. October 29, 2018 DRAFT SUBMITTED TO IEEE TRANS. INFORM. THEOR Y , AP R. 2008 10 W e term the resulting codes as Structured-La ST (S-LaST) code s. The third property yields goo d s haping gain γ s , d efined a s the ratio of the normalized sec ond moment of an n -dimensiona l hypercu be to that of the shaping region R . If the shaping region is an n -dimensional hy percube, as in the case of perfect codes, then γ s = 1 . Choosing a better shap ing region R doe s not change the geometric arrange ment of the la ttice points, but the av erage transmitted energy is decreased thank s to shaping. Th e above three requirement are simultaneou sly achiev ed using a nested lattice (V oronoi) c onstruction and a non-linear modulo-lattice en coder nickname d s phere enco der . 3 Let G p denote the generator matrix of a pe rfect code (un itary), and let G Λ denote the generator matrix of a good 2 M T - dimensiona l integral lattice Λ , that is, a lattice with large fundamen tal coding gain (s uch lattices are av ailable in the literature [6]). De fine Λ c to b e the lattice with ge nerator matrix G Λ c = G p G Λ and let Λ s (referred to a s the sha ping lattice) be a sublattice of Λ c such that Λ s has goo d shaping gain. Let [Λ c | Λ s ] deno te the nes ting ratio, that is, the cardinality o f the quotient group Λ c / Λ s . Then, we construct a s tructured LaST c ode X as the set of all distinct points x giv en by x = [ λ + u 0 ] mo d Λ s as λ v aries in Λ c , and u 0 is a trans lation vector used to symmetrize the cod e. Although not nece ssary , in all ca ses con sidered in this paper we let Λ s = Q Λ c , Q ∈ Z + for simplicity , i.e., we use a self-similar sha ping lattice. The rationale be hind this choice is that it is well-known that for mode rate d imensions, the best lattices with respect to coding gain are also good qu antizers, i.e., h av e good s haping gain. The c oding rate is gi ven by R = 1 T log[Λ c | Λ s ] = 2 M log Q . No tice a lso tha t bec ause of the “rotation” matrix G p and the fact tha t Λ is an inte gral lattice, the set of points X represented a s complex matrices has the NVD prope rty . Theorem 1 : The space -time code X derived from the lattice G Λ c = G p G Λ using a nes ted-lattice structure correspond s to a spa ce-time code derived from CD A with non -v anishing d eterminant and h ence achieves the op timal DMT over any fading chan nel statistics. Pr o of: Recall that G p correspond s to a ST code with NVD, i.e., the set of all non-zero lattice vectors z ∈ G p Z 2 M T , represe nted as complex matrices Z c , have det  Z c ( Z c ) H  bounde d away from ze ro by so me constan t term SNR 0 (up to order o f exponent of SNR). S ince Λ is an integral lattice, there exists 3 T ree-search algorithms to perform the C losest L attice Point S earch (CLP S), based on Pohst enumeration [26] and generalized in [22], [23] are generally nicknamed “sphere decoders” if used for minimum distance latti ce decoding or “sphere encoders” if used for modulo-lattice precoding, i n the current communication and coding theoretic l iterature. The reason of t he nickname follo ws from the bounded-distance enumerativ e decoding of the Pohst lattice point enumeration and v ariants t hereof. October 29, 2018 DRAFT SUBMITTED TO IEEE TRANS. INFORM. THEOR Y , AP R. 2008 11 −20 0 20 −15 −10 −5 0 5 10 15 −10 0 10 −15 −10 −5 0 5 10 15 Fig. 1. Illustrating the Sphere-Encoder: Hexagonal Latt ice, Q = 16 , linear map (left) and sphere-encod ed map (right) a k ∈ R such that k G Λ generates a subla ttice of Z 2 M T . It follows that the La ST code k X ge nerated by k G p G Λ is a sublattice of G p Z 2 M T and therefore s atisfies min X ∈ X : X 6 =0 det ( XX H ) ˙ ≥ k − 2 M SNR 0 . = SNR 0 . The proof of DMT op timality now follows from [2], [3]. The mo dulo- Λ s “sphere -encoder” is easily impleme nted by some CLPS, using so me “s phere de coding” algorithm [22 ], [23 ]. The sha ping effect of sphere-encoding is best ill ustrated using a 2 -dimensional example. Su ppose that Λ c is the h exagonal la ttice in two dimen sions. Set Q = 16 . The constellations correspond ing to the linear map (centred at the origin) an d the s phere-enco der are s hown in Fig. 1. A s the value of Q increases, the sphere-en coded c onstellation fills the fun damental V oronoi region of the hexagonal lattice uniformly . Althoug h both con stellations correspon d to signalling from the hexagon al lattice, the energy saving of the sphere-e ncoder is evident. Example 1: (The Golden-Gosset S-LaST co de) When M = 2 , we choose G p to be the lattice generator matrix of the Golden cod e [11] and G Λ to be the genera tor matrix of the Go sset lattice E 8 , which are respectiv ely gi ven by G p = 1 √ 5   Re ( G c p ) − Im ( G c p ) Im ( G c p ) Re ( G c p )   , October 29, 2018 DRAFT SUBMITTED TO IEEE TRANS. INFORM. THEOR Y , AP R. 2008 12 where G c p =         η θ η 0 0 0 0 η θ η 0 0 γ σ ( η ) γ σ ( θ ) σ ( η ) γ σ ( η ) γ σ ( θ ) σ ( η ) 0 0         , θ = 1+ √ 5 2 , σ ( θ ) = 1 − θ , η = 1 + ı − ıθ , σ ( η ) = 1 + ı − ıσ ( θ ) , γ = ı , and G Λ =                     2 − 1 0 0 0 0 0 0 . 5 0 1 − 1 0 0 0 0 0 . 5 0 0 1 − 1 0 0 0 0 . 5 0 0 0 1 − 1 0 0 0 . 5 0 0 0 0 1 − 1 0 0 . 5 0 0 0 0 0 1 − 1 0 . 5 0 0 0 0 0 0 1 0 . 5 0 0 0 0 0 0 0 0 . 5                     . Example 2: (The Golden+ Algebra ( GA + ) S-LaST code) Our second example is bas ed on a 2 × 2 ST code de ri ved from a maximal order o f a CD A [12]. Th e Golden+ a lgebra [12] is defined to be GA + = ( Q ( δ ) / Q ( ı ) , σ , ı ) , where δ is the fi rst qua drant square root o f 2 + ı and the a utomorphism σ is determined by σ ( δ ) = − δ . The max imal order O of GA + is ge nerated by t he follo wing orde red Z ( ı ) − basis:      1 0 0 1   ,   0 1 ı 0   , 1 2   ı + ıδ ı − δ − 1 + ıδ ı − ıδ   , 1 2   − 1 − ıδ ı + ıδ − 1 + δ − 1 + ıδ      . (8) The Golden + code [12] corresponds to the left ideal of the maximal order generated by M =   (1 − δ ) 3 0 0 (1 + δ ) 3   . (9) In this ca se, we choose G Λ to be t he lattice ge nerator matrix co rresponding to this left ide al of the maximal order and G p = I (tri vial rotation). Notice that this choice does not maximize the funda mental coding gain (the Golden-Gosset S-LaST cod e has a h igher d ensity), b ut the minimum determinant of the Golden + S-LaST code is better tha n that of the Golden-Gos set code. It is a priori not c lear which eff ect will do minate the performance in terms of error proba bility; this will be a nswered in the simulation results to follow . October 29, 2018 DRAFT SUBMITTED TO IEEE TRANS. INFORM. THEOR Y , AP R. 2008 13 C. P erforman ce under low-complexity MMSE-GDFE Lattice Dec oding Unfortunately , d ue to the us age of a non -linear encod ing to ac hieve shaping g ain, ML decoding of the resulting code is very complicated, requiring ess entially the e xhaus ti ve enume ration of the whole codebo ok. Notice that a s imilar problem a rises in the case of the GA + code in [12], where linear encoding would result in very bad shaping. T he authors in [12] have obtained shaping by enumerating the minimum energy co dew ords an d perform exhausti ve decoding, both these are feasible only for low spectral efficiencies. Hence, w e reso rt to suboptimal MMSE-GDFE lattice dec oding (see [1], [22] for deta ils). It ha s b een proven that this decoder a chieves the optimal DMT in the ran dom coding s ense, for a sp ecific e nsemble of ran dom lattices . H ere, we use it with our deterministic non-rando m constructions. W e do not claim that the resulting sch emes ach iev e the op timal DMT un der lattice decod ing. Nevertheless, the performance o f these codes is ou tstanding. In our simulations, we make use of a ran dom translation vector u 0 , u niformly distrib uted over a very large h ypercube with v olume much larger tha n the volume of the s haping region. This rando m “dithering” is known to the rec eiv er , an d is s ubtracted be fore d ecoding, as explained in [1]. W ith this “trick”, we ens ure tha t the transmitted points ha ve energy exac tly equal to the sec ond mome nt of Λ s and have exac tly z ero mean. Furthermore, d ithering symmetrizes the s cheme and makes the error probability indep endent of the transmitted codeword. Fig. 2 compares the pe rformance of two 2 × 2 ST c odes deriv ed from CD A with R = 16 bpcu and N = 2 . The two ST codes ch osen in this case have γ c (Λ c ) equal to 0 . 8365 and 1 . 4142 respectiv ely . Sphere enco ding and MMSE-GDFE lattice de coding are used in both cases . W e notice about one dB of gain due to better funda mental coding gain of the lattice. In order to illustrate the benefit of cons tellation shaping , w e plot in Fig. 3 the p erformance of a ( 2 × 2 ) ST code deriv ed from C D A fi rst using linea r encod ing of the information symb ols a nd ML decoding and then using sphe re e ncoding and MMSE-GDFE decoding ( R = 16 bpcu, N = 2 ). The particular ST c ode chosen has γ c (Λ c ) = 0 . 8365 . Quite a significan t g ain of about 3 . 5 dB resu lts from codebook shap ing in this particular case. For the ca se o f M = 2 , we c ompare the performance of the Golden C ode [11], which is a perfect 2 × 2 ST cod e (with γ c (Λ c ) = 1 ), with the Golden-Gosset 2 × 2 S-LaST code from Example 1, ( γ c ( E 8 ) = 2 ). Fig. 4 shows plots of the Golde n c ode unde r ML dec oding a nd MMSE-GDFE latti ce decoding in comparison with the Golde n-Gosset S-LaST code with MMSE-GDFE lattice deco ding at rates of 4 and 16 bpc u. At 4 b pcu, the (real) information symbol co nstellation co rresponds to BPSK signaling on October 29, 2018 DRAFT SUBMITTED TO IEEE TRANS. INFORM. THEOR Y , AP R. 2008 14 24 26 28 30 32 34 36 38 40 42 10 −4 10 −3 10 −2 10 −1 10 0 SNR (dB) Codeword Error Prob. Fundamental Coding Gain = 0.8365 Fundamental Coding Gain = 1.4142 Outage Probability, 16 bpcu Fig. 2. E ffec t of fundamental coding gain on performance: 2 × 2 S T codes deriv ed f rom CDA, 16 bpcu, N = 2 , MMSE-GDF E lattice decoding 24 26 28 30 32 34 36 38 40 42 10 −4 10 −3 10 −2 10 −1 10 0 SNR (dB) Codeword Error Prob. (2 × 2) CDA code, Linear map, ML Decoding (2 × 2) CDA code, Sphere Encoded map, MMSE−GDFE Lattice Decoding Outage Probability, 16 bpcu Fig. 3. Effect of shaping gain on performance: 2 × 2 ST code deriv ed from CDA, 16 bpcu, N = 2 October 29, 2018 DRAFT SUBMITTED TO IEEE TRANS. INFORM. THEOR Y , AP R. 2008 15 5 10 15 20 25 30 35 40 45 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 SNR (dB) Codeword Error Prob. Golden Code, 4 bpcu, MMSE−GDFE Gosset ST Code, 4 bpcu, MMSE−GDFE Golden Code, 4 bpcu, ML Decoding Golden Code, 16 bpcu, MMSE−GDFE Gosset ST Code, 16 bpcu, MMSE−GDFE Golden Code, 16 bpcu, ML Decoding Outage, 16 bpcu Outage, 4 bpcu Fig. 4. Comparing the Golden Code with the Rotated Gosset Latti ce ST Code, N = 2 each dimension ( Q = 2 ). In this c ase, the sign al points of the Golden cod e in 8 -dimensional sp ace lie on the surface of a sphere (they are v ertices of t he rotated hypercu be). Therefore, the 2 × 2 perfect code construction is optimal for 4 bpcu also in terms of s haping. This intuiti on is verified by the plots correspond ing to 4 bpcu in Fig. 4. Howe ver , when the numbe r o f bits per chann el use increase s, the effect of the coding gain of the lattice an d the shaping gain begin to show up. At 16 b pcu, the Golden-Goss et S-LaST c ode with MMSE-GDFE lattice deco ding (mar ginally) outperforms the Go lden co de with ML decoding (se e Fig. 4 ). The se plots als o serve to illustrate that MMSE-GDFE lattice d ecoding is near -ML in performanc e, while offering significant reductions in complexity . In Fig. 5, we present c omparisons of the Go lden code with ML d ecoding, the Golden-Gosse t S-LaST code (see Exa mple 1) and the GA + S-LaST code (see Exa mple 2), at 16 bpcu. While the fun damental coding gain o f the lattice co rresponding to the G A + code is les s tha n the coding g ain of E 8 , the loss in density is compens ated for by an increase in the minimum determinant. Both the Go lden-Gosse t an d the GA + S-LaST code s with MMSE-GDFE lattice deco ding outperform the Go lden code with ML de coding. For the 3 × 3 ca se, we comp are the performanc e of two pe rfect codes from [5] and [4] (with base alphabets QAM and HEX respectively) with a n S-LaS T code b ased o n a rotated version of the Λ 18 lattice, which is the best kno wn lattice pac king in 18 -dimensions [6]. MMSE-GDFE lattice decoding is u sed for all cases . Th e results shown in Fig. 6 show a sign ificant gain for both 6 and 24 bpcu resulting from the October 29, 2018 DRAFT SUBMITTED TO IEEE TRANS. INFORM. THEOR Y , AP R. 2008 16 Fig. 5. Performanc e of the 2 × 2 Golden code, Golden-Gosset and G A + S-LaST codes at R = 16 bpcu. The inset sho ws a portion of the plot zoomed for clarity . increased lattice coding gain and s haping. In Fig. 7 we compare the perform ance of the 2 × 2 Golden-Gosse t S-La ST code ( T = 2 ) with rectangular 2 × 4 and 2 × 6 S-LaST codes constructed us ing the horizontal-stacking con struction [2] in con junction with the Ba rnes-W all ( Λ 16 ) ( γ c (Λ 16 ) = 2 . 8284 ) and Leec h ( Λ 24 ) ( γ c (Λ 24 ) = 4 ) lattices respecti vely . The length- 24 cyclic code G 24 ( Z 4 ) constructed in [10] was u sed to cons truct an isomorphic version of the Leech lattice using construction-A [6]. MMSE-GDFE latt ice de coding is used for all three S T codes. In accordan ce with intuition, the performanc e approach es o utage probab ility as T i ncreas es, owing to better values of γ c (Λ c ) . I V . T H E S - L A S T T C M S C H E M E Moti vated by the fact that in practical wireless co mmunications M is limited by transmitter complexity to be a s mall integer (typically 2 or 4, in current IEEE802.11n MIMO extens ion of wireless local area networks) while T may be of the order of 1 00 channel uses, our objecti ve in this sec tion is to c onstruct M × T S T cod es for the case of T ≫ M . F or eas e of expo sition and without loss of fundame ntal generality , we will focu s on the case whe re T = LM , for s ome integer L . TCM h as the nice fea ture that a single trellis code can ge nerate any desired block length, with decoding complexity linear in L , using October 29, 2018 DRAFT SUBMITTED TO IEEE TRANS. INFORM. THEOR Y , AP R. 2008 17 5 10 15 20 25 30 35 40 45 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 SNR (dB) Codeword Error Prob. 3*3 Perfect Code (Elia et al.), 6 bpcu 3*3 Perfect Code (Oggier et al.), 6 bpcu 3*3 Rotated Λ 18 Lattice, 6 bpcu 3*3 Perfect Code (Elia et. al.), 24 bpcu 3*3 Perfect Code (Oggier et. al.), 24 bpcu 3*3 Rotated Λ 18 Lattice, 24 bpcu 3*3 MIMO Outage, 6 bpcu 3*3 MIMO Outage, 24 bpcu Fig. 6. 3 × 3 ST Codes under MMSE-GDFE lat tice decoding, N = 3 33 34 35 36 37 38 39 40 41 42 10 −3 10 −2 SNR (dB) Codeword Error Prob. Rotated Gosset Lattice, T = 2 Rotated BW−16 Lattice, T = 4 Outage (16 bpcu) Rotated G 24 (Z 4 ) lattice, T = 6 Fig. 7. Increasing the Coding Length, M = N = 2 , T = 2 , 4 , 6 , R = 16 bpcu, MMSE-GDF E lattice Decoding October 29, 2018 DRAFT SUBMITTED TO IEEE TRANS. INFORM. THEOR Y , AP R. 2008 18 Fig. 8. S-LaST T CM Encoder a V iterbi deco der . Furthermore, the co nstruction of TCM sc hemes is rather well unde rstood and a rich literature exists for the Gaus sian cha nnel (se e [13], [14] , [15] and references therein), the scalar f ading channe l (see [16] and references therein) and for the MIMO f ading c hannel [17], [18], [19]. A. En coder Consider a three level p artition Λ t ⊃ Λ m ⊃ Λ b (where the su bscripts indicate ‘top’, ‘middle’ and ‘bottom’) o f lattices in R n , with n = 2 M 2 . Let [Λ t | Λ m ] = M and let the co sets of Λ m in Λ t be indica ted by C i , { v i + Λ m } , for i = 1 , . . . , M , whe re eac h v i is a cose t repres entativ e of C i . From ea ch c oset C i , we carve a finite set of N points, d enoted by { v i + c j : c j ∈ Λ m , j = 1 , . . . , N } . The se po ints a re c hosen via a modulo- Λ b sphere e ncoder , that will be des cribed in the follo wing . Also, we choose Λ b such that N = [Λ m | Λ b ] . In all the examples presented here, we use Λ b = Q Λ m , for so me Q ∈ Z + (i.e., we us e again a s elf-similar shap ing lattice). In this case, N = Q 2 M 2 . W e make use of Forney’ s g eneral “co set coding ” frame work [8]. A block diag ram of t he encoder is s hown in Fig. 8. During e ach b lock k = 1 , . . . , L c omprising of M ch annel use s eac h, a block of (log M ) /r + log N information bits enters the e ncode r . Th e top (log M ) /r information bits are input to a c on volutional encoder of (binary) rate r , that outputs log M coded b its, wh ich select the index i k ∈ { 1 , . . . , M } of a c oset in Λ t / Λ m . Th e remaining log N information bits select the ind ex j k of a point in the finite con stellation carved from the s elected cose t C i k . October 29, 2018 DRAFT SUBMITTED TO IEEE TRANS. INFORM. THEOR Y , AP R. 2008 19 The transmitted vector at time k is gi ven by x k = [ c j k + v i k + u k ] mo d Λ b (10) where u k is an optiona l rand om d ithering signal known to the receiv er , that se rves to symme trize the overall TCM code and to induc e the uniform error property . The vector x k is then mappe d into an M × M complex matrix and transmitted in M channel u ses across the MIMO channel. The rate of the S-LaST TCM sc heme is given b y R = (log M ) /r + log N M bits/channel use . It sh ould be noticed tha t x k = c j k + v i k + u k − λ k for some λ k ∈ Λ b that is a function of c j k , v i k , u k . Further , x k ∈ V (Λ b ) . Since [Λ m | Λ b ] = N , the map ping between the uncode d bits an d the constellation points in e ach co set is one-to-one . B. De coder The (real equiv a lent) recei ved p oint at each block k is gi ven by y k = Hx k + w k , for k = 1 , . . . , L . In gene ral, the trellis of the S-LaST TCM scheme has N parallel transitions p er trelli s branch, correspon ding to the N points in the interse ction C i ∩ V (Λ b ) , on e ach branch labeled by the coset C i . Co nsider time k , and a branch labe led by coset C i . Th e corresp onding branch me tric for a ML trell is decode r (impl emented via the V iterbi algorithm) is given by B i,k = min c ∈ Λ m ∩ V (Λ b ) | y k − H ( v i + c + u k ) | 2 . (11) Computing this branch metric amounts to exhaus ti ve enume ration of all p oints of Λ m in the V orono i region V (Λ b ) of the shaping lattice. Since exhaus ti ve enumeration is usu ally too complex, we resort once ag ain to a suboptimal MMSE- GDFE lattice de coder along the lines of [1], in order to compu te an approximate ML bran ch metric for the V iterbi decod er . First, we relax the minimization in (11 ) to take into ac count all points of Λ m (Lattice decoding ), i.e., we con sider the sub optimal branch metric B i,k = min c ∈ Λ m | y k − H ( v i + c + u k ) | 2 . (12) This amou nt to solving a CLPS p roblem for the channel-modified lattice H Λ m , with res pect to the point y k − H ( v i + u k ) , w here u k is a known dithering vector an d v i depend s on the label of the branch for October 29, 2018 DRAFT SUBMITTED TO IEEE TRANS. INFORM. THEOR Y , AP R. 2008 20 which we c ompute the metric. The surviving path among the pa rallel p aths corresponds to the argument c that minimizes (12). Then, we further modify the suboptimal metric follo wing the MMSE-GDFE paradigm (see [1] for the details). Let F and B de note the forward and backward filters of the MMSE-GDFE as defin ed in [1]. At eac h time k , the receiver ob tains the following set of mo dified cha nnel observations y ′ i,k = Fy k − B ( v i + u k ) , 1 ≤ i ≤ M . Using the properties of the ma trices F and B , thes e can be written as y ′ i,k = F [ H ( c j k + v i k + u k − λ k ) + w k ] − B [ u k + v i ] = B ( c j k + v j k − λ k − v i ) − [ B − FH ]( c j k + v i k − λ k + u k ) + Fw k = B ( c j k + v j k − λ k − v i ) − [ B − FH ] x k + Fw k , B ( c j k + v j k − λ k − v ℓ ) + e ′ k . Notice that x k is uniformly distrib uted over V (Λ b ) and is h ence independen t of c j k and v j k [1]. It c an be shown that the noise plus self-noise vector e ′ k has the sa me covariance matrix of the original n oise w k , although it is ge nerally non-Gaussian . Also , v i k − v i = 0 (i.e., it be longs to Λ m ) if i k = i , while it belongs to some coset of Λ m in Λ t not eq ual to Λ m if i k 6 = i . For each branc h labeled by c oset C i , the low-complexity V iterbi decod er computes branch me tric B i,k = min z ∈ Z 2 M 2   y ′ i,k − BG Λ m z   2 where G Λ m denotes a generator matrix for Λ m . Th is c an be obtained by a sph ere decode r ap plied to the channe l-modified lattice B Λ m . It is clea r that the branc h metric for the c orrect c oset (i.e., for i = i k ) will be s maller than the branch metric for an incorrect coset, with high probability . C. Con struction of su itable lattice p artition chains In o rder to en sure good p erformance, we choos e the compon ent M × M code of the S-La ST TCM scheme to be approximately un i versal. W e will therefore choose Λ t to be the lattice corresponding to an ST code derived from CD A with NVD. In order to co nstruct Λ m and Λ b , we will first discu ss the important spe cial case when Λ t correspond s to a perfec t code , and then treat the more gene ral case. October 29, 2018 DRAFT SUBMITTED TO IEEE TRANS. INFORM. THEOR Y , AP R. 2008 21 1) P artitions of perfect code s: Let Λ t be the lattice corresponding to a perfect c ode [4], [5], with generator matrix G p . Then, Λ t is a rotated version o f the cub ic lattice Z 2 M 2 . Following wha t was done before for the cas e of short block co des, we ch oose Λ m to be the bes t known integral lattice packing in 2 M 2 − dimensiona l space, rotated b y G p . Also, we s et Λ b = Q Λ m . For example, when M = 2 , we choose Λ m to b e the Golden Gosse t lattice. The resulting code sha ll be name d the Golde n-Gosset S-LaS T TCM sc heme. 2) S-LaST TCM fr om max imal orders in CD A s: W e choos e Λ t to b e the latti ce correspon ding to the maximal order of a giv en CD A. An exa mple for the c ase whe n M = 2 would be the lattice c orresponding to the GA + code that we made use of for the short block-length case in Example 2. Simil ar to the approach used in [20], [21] for the cubic lattice c ase, we will use ideals β O o f the maximal order for the sublattice Λ m . The element β yielding a good sublattice is obtained through a c omputer search, that makes us e o f the following lemma. Lemma 2: Let D ( L / Q ( ı ) , σ, γ ) be a cyclic d i vision alge bra of index n , and let O denote an orde r of D . If β is an element of the orde r , then [ O | β O ] = | N r ( β ) n | 2 . Pr o of: Althoug h this lemma is well known to the mathematics co mmunity , we provide a sketch of the proof for c ompleteness . Con sider any β ∈ O . The n β indu ces a transformation on O with image β O . These are finitely generated fr ee modules over Z , and s o the index of partition is just the d eterminant o f β in this action. W e may compute the determinant over the c orresponding fie ld. D has rank 2 n 2 over Q . First viewi ng D a s a (right) vector sp ace o f dimension n 2 over Q ( ı ) , we see that the determinant of multiplication by β is N r ( β ) n . W e then apply the norm from Q ( ı ) to Q to obtain the determinant. The comp uter search performs the followi ng: 1) Fix a desired index of partition M = [Λ t | Λ m ] , and a sufficiently lar g e integer ν . 2) Let O ν denote the integral closure of {− ν , − ν + 1 , . . . , ν − 1 , ν } ⊂ Z in O . More specifically , if γ 1 , γ 2 , . . . , γ 2 M 2 constitutes a b asis for O ov er Z , then O ν , ( 2 M 2 X i =1 g i γ i      − ν ≤ g i ≤ ν, g i ∈ Z ∀ i ) . Notice that such a bas is alw ays exists, since e very a lgebraic number field has at lea st one integral basis [25]. 3) F or e ach β ∈ O ν that g enerates a pa rtition with required index M , i.e., satisfying   N r ( β ) M   2 = M , October 29, 2018 DRAFT SUBMITTED TO IEEE TRANS. INFORM. THEOR Y , AP R. 2008 22 compute the funda mental co ding ga in of the lattice corresp onding to β O , and let β max denote a maximizer . 4) Set Λ m to be the lattice correspon ding to β max O . Finally , as b efore, we us e the self-similar shaping lattice Λ b = Q Λ m , for s ome Q ∈ Z + . D. Cod e cons truction examples In this se ction, we present two cons truction example s of S-LaST TCM, the p erformances of w hich are compared by s imulation. • The Golden-Gosset S-LaST TCM cons truction (se e Example 1 ): h ere Λ t = G p Z 8 , Λ m = G p E 8 and Λ b = Q Λ m , Q ∈ Z + . • The GA + S-LaST TCM cons truction: we c hoose Λ t to be the lattice correspon ding to the GA + S- LaST cod e in Example 2. Λ m is obtaine d using the co mputer search giv en above, a nd correspo nds to the left i deal of β 2 O g enerated by M (gi ven in (9)), whe re O is the maximal o rder of t he GA + algebra (see Example 2) a nd the coo rdinates of β in terms of the ordered b asis in (8) are ( − 1 , − 1 , 1 − ı, − 1 − ı ) . W e then set Λ b = Q Λ m , Q ∈ Z + . Both these c odes co rrespond to a 16 − ary pa rtition Λ t / Λ m , as s hown in Fig. 9. The minimum de terminant Fig. 9. T wo l e vel partition of the example constructions increases a s one goes down the partiti on chain. W e us e the trellis shown in F ig. 10 that is designe d s uch that the transitions leaving/mer g ing into a state have ma ximum pos sible minimum determinant. In our simulations, we hav e used block length T = 260 chan nel uses, corresponding t o 1300 information bits per packet, at R = 5 b pcu. F ig. 11 shows the pe rformance in te rms of pac ket error probability of the October 29, 2018 DRAFT SUBMITTED TO IEEE TRANS. INFORM. THEOR Y , AP R. 2008 23 Fig. 10. 16-state trell is used for the example constructions above two S-LaST TCM sch emes in comparison with the Golden ST TCM (GST -TCM) scheme [21] at 5 bpcu. Also sh own is the performance of the “uncoded Golden code ” cons truction [21], which cons ists of stacking 130 Golden code matrices next to each other (coding is performed only over 2 time-slots). The propos ed S-LaST TCM construction is see n to gain arou nd 1 dB over the GST -TCM sche me. V . C O N C L U S I O N S In this pa per , we hav e ad vocated the use of structured lattices that are e ndowed with good pac king and shap ing prope rties in the d esign o f spac e-time co des with both sho rt and long block -lengths. Th e constructions p resented have reaso nable dec oding c omplexity , an d exhibit excellent pe rformance in terms of error p robability . Quite a few res earch topics occ ur naturally as potential follow-up works. While codes with s hort b lock- length have p erformances that are very c lose to the outag e probability , the re is s till quite a signific ant gap from outage for the case of long block -lengths. Designing better codes for this scena rio remains a challenging open problem. It would also be interesting to explore if there exist better algebraic frame works that allow us to choose sublattices with g ood pac king and sha ping properties. October 29, 2018 DRAFT SUBMITTED TO IEEE TRANS. INFORM. THEOR Y , AP R. 2008 24 4 6 8 10 12 14 16 18 20 22 10 −3 10 −2 10 −1 10 0 E b /N 0 (dB) P e Golden ST−TCM, 16 states Gosset S−LaST TCM, 16 states Uncoded Golden Code Golden+ S−LaST TCM Outage, 5 bpcu Fig. 11. Performance of the Golden-Gosset and G A + S-L aST TCM schemes, R = 5 bpcu, T = 260 A C K N O W L E D G E M E N T S The autho rs would like to thank Prof. Robert Gura lnick for s ome useful d iscussions . R E F E R E N C E S [1] H. E l Gamal, G. Caire and M.O. Da men, “Lattice Coding and De coding Achiev e the Optimal Di versity-Multilpexing T radeoff of MIMO Channels, ” IEEE Tr ans. Inform. Theory , V ol. 50, No. 6, pp. 968-985, June 2004. [2] Petros Elia, K. Raj Kumar , Sameer A. Pawar , P . V ijay Kumar and Hsiao-feng L u, “Explicit, Minimum-Delay Space-T ime Codes Achie ving T he Diversity-Multiple xing Gain T radeof f, ” I EEE T rans. Inform. Theory , V ol. 52, No. 9, pp. 3869 - 3884, Sept. 2006. [3] S. T avildar and P . V iswanath, “ Approximately uni versal codes o ver slo w fading chan nels, ” IEEE T rans. Inform. Theory , V ol. 52, No. 7, pp. 3233 - 3258, Jul. 2006. [4] F . Oggier , G. Rekaya, J -C. Belfiore and E. V iterbo, “Perfect S pace T ime Block Codes, ” IEEE T rans. Inform. Theory , V ol. 52, No. 9, pp. 3885-3902, Sept. 2006. [5] P . E lia, B. A. S ethuraman and P . Vijay Kumar ,“Perfect Space-Time Codes with minimum and non-minimum delay for any number of transmit antennas”, IEE E T rans. Info. T heory , V ol. 53, No. 11, pp. 3853-3868, Nov . 2007. [6] J. H. Conway and N . J. A . Sloane, Spher e P ackings, Latti ces and Gro ups , Second Edition, Springer-V erlag, 1993. [7] Gabriele Nebe and N. J. A. S loane, A Catalogue of Lattices , ht tp://www.rese arch.att.com/ ∼ njas/lattices/ index.html . [8] G. D. Forne y , Jr ., “Coset Codes-Part I: Introduction and Geometrical Classification, ” IEEE T rans. Info. Theory , V ol. 34, No. 5, pp. 1123-1151, Sept. 1998. [9] L. Zheng and D. Tse, “Div ersity and Multiplexing: A Fundamental Tradeof f in Multiple-Antenna Channels, ” IEEE T ran s. Inform. Theory , V ol. 49, No. 5, pp. 1073-1096, May 2003. October 29, 2018 DRAFT SUBMITTED TO IEEE TRANS. INFORM. THEOR Y , AP R. 2008 25 [10] G. Caire and E. Biglieri, “Linear block codes ov er cyclic grou ps, ” IEEE T rans. Inform. Theory , V ol. 41, No. 5, pp. 1246-1256, Sep. 1995. [11] J.- C. Belfiore, G. R ekaya and E . V iterbo, “The Golden code: a 2 × 2 full-rate space-time code with non-vanishin g determinants, ” IEE E T rans. Inform. T heory , V ol. 51, No. 4, pp. 1432-1436, Apr . 2005. [12] C. Hollanti, J. Lahtonen, K. Ranto and R. V ehkalah ti, “On the dense st MIMO lattices from cyclic div ision algebras, ” Submitted to IEE E T rans. Inform. Theory , Dec. 06. [13] G. Ungerboeck, “Chann el coding with multile vel/pha se signals, ” IEE E T rans. Inform. Theory , V ol. 28, No. 1, pp. 55-67, Jan. 1982. [14] G. D. Forney , Jr. and G. Ungerboeck, “Modulation and coding for linear Gaussian channels, ” IEEE T rans. Inform. T heory , V ol. 44, No. 6, pp. 2384-2415, Oct. 1998. [15] C. Schlegel, T r ellis Coding , IEEE P ress, New Y ork, 1997. [16] S . H. Jamali and T . Le-Ngoc, Coded-Modulation T echn iques for F ading Channels , Kluwer Academic Publishers, Massachusetts, 1994. [17] V . T arokh, N. Seshadri, and A. R. Calderbank, “Space-time codes for high data rate wireless communication: performance criterion and code construction, ” I EEE T rans. Inform. Theory , V ol. 44, No. 2, pp. 744-765, Mar . 1998. [18] S . Siwamogsatham and M. P . Fitz, “High-Rate Concatenated SpaceTime Block Code M-TCM Designs, ” IE EE T rans. Inform. Theory , V ol. 51, No. 12, pp. 4173-4183 , Dec. 2005. [19] H. Jafarkha ni and N. S eshadri, “Super-Orthogon al S paceT ime T rellis Codes, ” I EEE T rans. Inform. Theory , V ol. 49, No. 4, pp. 937-950, Apr . 2003. [20] D. Champion, J.-C. Belfiore, G. R ekaya and E . V iterbo, “Partitionning the Golden Code: A frame work to the design of Space-T ime coded modulation, ” Canadian W orkshop on I nfo. Th. , Montreal, 2005. [21] Y i Hong, E. V iterbo and J-C. Belfiore, “Golden S pace-T ime T r ellis Coded Modulation, ” IEEE T rans. Inform. T heory , V ol. 53, No. 5, pp. 1689-1705, May 2007. [22] M. O. Damen, H. El Gamal and G. Caire, “On maximum l ikelihood detection and t he search for the cl osest lattice point, ” IEEE T rans. Inform. Theory , V ol. 49, No. 10, pp. 2389-2402, Oct. 2003. [23] A. D. Murugan, H. El G amal, M. O. Damen, and G. Caire, “ A U nified Framew ork for Tree Search Decoding: Rediscov ering the Sequential Decoder , ” IEEE Tr ans. Inform. Theory , V ol. 52, No. 3, pp. 933-953, Mar 2006. [24] B. A. Sethuraman, B. Sundar Rajan and V . S hashidhar , “Full-div ersity , High-rate, S pace-T ime Block Codes from Div ision Algebras, ” IEEE Tr ans. Info. Theory , V ol. 49, No. 10, pp. 2596-2616, Oct. 2003. [25] Harry Pollard and Harold G. Diamond, The Theory of Algebr aic Numbers , Third, Re vised Editi on, Dov er P ublications Inc., New Y ork, 1998. [26] E . Viterbo and J. Boutros, “ A Univ ersal Lattice Code Decoder for Fading Channels”, IE EE T ransactions on Information Theory , V ol. 45, No. 5, pp. 1639-1642 , July 1999. October 29, 2018 DRAFT