Space Time Codes from Permutation Codes

Space T ime Codes from Pe rmutation Codes Oli ver Henkel Fraunhofer German -Sino Lab for Mobile Commun ications - MCI Einsteinufer 37, 10587 Berlin, Germany Email: henkel@h hi.fraunhof er.de Abstract — A new class of space time codes with high per - fo rmance is prese nted. The code design util izes tailor -made permutation codes, which are known to h a ve lar ge minimal distances as spherical codes. A geometric connection b etween spherical and space time codes has been used to translate them into the final space time codes. Simu lations demonstrate t hat the performance increases wi th the block lengths, a result that has been conjectured already in pre vious work. Further , the connection to permutation codes allows for moderate complex en-/decoding algorithms. I . I N T R O D U C T I O N In MIMO (Multip le In put Multip le Output) systems space time coding schemes have been proven to be an app ropriate tool to exploit the spatial diversity gains. T wo distinct scenar- ios are com mon, wheth er th e chann el coefficients are known (cohere nt scenario) [1], to the receiver or not (n on-coh erent scenario) [2]. Pro minent coh erent cod es are the well known Alamouti scheme [ 3] and general or thogon al design s [ 4]. A more flexible coding scheme are the so-called linear dispersion codes. They have been introduc ed in [5] and w ere fu rther in vestigated in [6]. A high rate example achieving the di versity multiplexing tradeo ff is the recently discovered Gold en code [7]. Genuine no n-coher ent codes have been p roposed in [8], but most of the re search efforts in th e literature focu s on differential schemes, introdu ced in [9], since differential codes usually provide higher data rates than comp arable no n d iffer - ential co des. High per forming examples hav e been con structed in [10], [11],[1 2],[13]. However , in both cases most re search effort h as b een under taken for space time blo ck codes with quadra tic 2-b y-2, resp . n t -by- n t code matrices ( n t denotes the number o f tr ansmit anten nas). Altho ugh linear dispersion codes are no t restricted to quadr atic shap e of the design matrices the block leng th is n ot a free desig n p arameter when the num ber of transmit anten nas is held fixed (comp are th e asymptotic gu idelines in [6]). In contrast to that, both coher ent and (n on-d ifferential) non- coheren t case are expected to b enefit from cod ing schem es which use the addition al degrees o f f reedom provided by in- creasing the block length [14] (whereas n t is fixed). This result has or iginally been developed in the context of packin g theory , but in [ 15] its influence on the per forman ce on sp ace time block code s h as been poin ted ou t. Roughly speaking, space time code design can be co nsidered as a con strained spher e packing problem, whe re the objective (per forman ce gain) can be optimized in a tw o stage pro cess. St ep one aims to construct good pack ings, step two is concern ed with the m aximization of the codin g gain, gi ven a packing configur ation. This meth od works for the coher ent scenario as well as for the non -coher ent system. The present work utilizes the pro posed two stage process to construct space time cod es for bo th scenarios. It tu rns ou t that the per formanc e in term s of bit err or rates of the construc ted codes inc reases with the block length, in acco rdance to what has been conjectu red in [1 4]. T he simulation results show , that it is possible to beat the p erform ance o f some o ptimal conv entional 2-by -2 sch emes considerab ly . The two optimization steps, tho ugh d ifferent in their na ture, are comm only form ulated in geo metric terms, accor ding to the underly ing geo metric structur es o f the coding spaces. While the second step is simp ly a suitably de fined rotation of the data (precod ing in some sense), th e first step in volves geometric and co mbinator ial aspects. Th e d ifferential g eometric aspects have been alread y analyzed in previous publications [14], [15], [16], [17], and th e co ntribution of this work has its fo cus on the combinato rial pa rt, nam ely the construction o f app ropriate spherical perm utation co des. Section II introd uces the channe l mod el and basic defini- tions, section III states the cod e design criteria with emp ha- sis on the aspects which b ecome important for the further development, in particu lar sub section III-C summar izes th e main p oints. Section IV sketches the r esults of previous work, namely the differential geo metric co nnection between spherical pa ckings — which occur e. g. in the context optimal sequence de sign in CDMA systems — and pack ings on the Stiefel and Grassmann m anifolds, the appr opriate c oding spaces for space time bloc k co de d esign. Then in sectio n V permutatio n c odes enter the stage, since they ca rry naturally the interpretation a s sphe rical p ackings. Th e d esign of permu - tation codes yieldin g large packing distance s o n spheres with prescribed dimension and rate requirem ents will b e in vesti- gated, follo wed i n VI by an analysis of the s econd optimization step, i.e. the design of an approp riate ro tation matrix . Section VII p resents simulations o f bit error perfor mance an d VI II summarizes th e work do ne so far, f ollowed by an outlook to further work. I I . C H A N N E L M O D E L A N D C O D I N G S PAC E S Let us assume a MIMO system with n t transmit anten nas and n r receive anten nas. The fading statistic is assumed to obey a Rayleigh flat fadin g mod el with block length T o f th e coheren ce inter val. Then we have the transmission equatio n Y = √ ρ X H + N (1) where X denotes the T -by- n t transmit signal with nor malized expected power per time step, H ∼ C N ( 0 , 1 ) is the n t -by- n r circular symmetric complex normal distrib u ted channel matrix, N ∼ C N ( 0 , 1 ) denotes the T -by- n r additive noise, and Y the T -by- n r received signal, whe re ρ tur ns out to be the SNR a t each receive an tenna. The symbol 1 de notes a u nit matrix throug hout this work, sometimes sup plemented by an index indicating the d imension. Due to the work of Hoch wald/Marzetta [18] it is reason able from a capacity perspective to assume the transmit signals X to have ( apart fro m a scaling factor) u nitary column s. M ore precisely we can wr ite X = r T n t Φ (2) and consider th e complex Stiefel manifo ld V C n t ,T := { Φ ∈ C T × n t | Φ ∗ Φ = 1 n t } (3) as the codin g spac e ( · ∗ denotes the hermitian conju gate). Thus a space tim e code is considered to be a d iscrete sub set C ⊂ V C n t ,T and we de fine the rate R of the co de b y R := 1 T log 2 |C | (4) Provided a re ceiv ed sig nal ˜ Y = q ρ T n t Ψ + N the m aximum likelihood (ML) detection r ule reads Φ ML = arg min ∀ Φ ∈C      ˜ Y − r ρ T n t Φ H      F (5) where k A k F = √ tr A ∗ A denotes the Fro benius norm. A. Non-coh er en t detectio n If the receiver h as n o inform ation abou t the fading states the detection is called non-co herent. In this case it is shown in [18], [2], [1 9] that the co ding spa ce is the complex Grassmann manifold G C n t ,T := {h Φ i | Φ ∈ V C n t ,T } (6) of n t -dimension al lin ear complex subspaces of C T ( h Φ i denotes the vector space span ned by the colum ns of the matrix Φ ). One can th ink o f Φ r epresenting a subsp ace h Φ i , but for a given Φ ∈ V C n t ,T all matrices Φ u with arbitrary unitar y n t -by- n t matrix rep resent the same sub space; th erefore the Grassmann manifold is really a coset space of th e Stiefel manifold and the choice of a un ique rep resentative f or each coset is no t obviou s in gener al. Howe ver , the maximu m likelihood d etection f or no n-coh erent detection d ecides on the subspace h Φ ML i represen ted b y Φ ML = arg max ∀ h Φ i∈C    ˜ Y ∗ Φ    F (7) giv en a ’ received n oisy subsp ace’ h ˜ Y i represented b y ˜ Y = q ρ T n t Ψ + N . Since the Frobe nius norm is unitarily inv ar iant, the ML criterion (7) is ind ependen t of the chosen represen- tati ves Φ an d Ψ , thus (7) provides a well defined measure of subspace correlation. Theref ore, the explicit c hoice o f a representative Φ o f h Φ i ∈ C is irrelev ant and we are f ree to consider non -coher ent codes C as subsets of the Stiefel manifold V C n t ,T rather than subsets of the Grassmann manifold, thinking in terms of repr esentativ es. As a notational con vention entities from a n on-coh erent context will be und erlined. I I I . S PAC E T I M E C O D E D E S I G N C R I T E R I A R E V I S I T E D A. Coher ent case: The code design aims to maximize an appropriate functional on the set of difference symbols ∆ := Φ − Ψ . Commo n design criteria arise fro m the familiar Chern ov bound for the pairwise error prob ability , which has the form [2] ch = 1 2 n t Y i =1  1 + σ 2 i (∆)  ! − n r (8) where  := 1 4 ρ T n t and σ ( A ) = ( σ i ( A )) g enerically denotes the vector of singu lar values o f a matrix A in d ecreasing orde r . T aking this bo und as the target function al it is immed iately clear that the cod e design do es not d epend on the num ber of receive antenn as, and the objective beco mes the maximization of the diversity fun ctional D i v := n t Y i =1  1 + σ 2 i (∆)  = n t X i =0 s i  i (9) where s j := sym j ( σ 2 1 (∆) , . . . , σ 2 n t (∆)) and sym j de- notes the j -th elementary sym metric polyn omial defined by sym j ( x 1 , . . . , x n t ) := P 1 ≤ i 1 ≤···≤ i j ≤ n t x i 1 · · · x i j . The diversity contains as its first order term the receiver metric itself, the so-called div ersity sum d 2 := s 1 = k ∆ k 2 F (10) as well as the diversity produ ct as its leading term p 2 := s n t = det(∆ ∗ ∆) (11) B. Non-coh er en t case: Follo wing [2] a similar deriv ation applies: Defining the codeword dif ference sym bol as ∆ := Φ ∗ Ψ the Che rnov bou nd now read s ch = 1 2 n t Y i =1  1 +  (1 − σ 2 i (∆))  − n r (12) where  :=  2  + 1 4 , and th e cor respondin g diversity qu antities become D i v := n t Y i =1  1 +  (1 − σ 2 i (∆))  = n t X i =0 s i  i (13) with s i := sym i  (1 − σ 2 1 (∆ )) , . . . , (1 − σ 2 n t (∆ ))  , and d 2 := s 1 = n t − k ∆ k 2 F (14) p 2 := s n t = det( 1 − ∆ ∗ ∆ ) (15) C. Implica tions for the cod e design a nd known results Coherent and n on-coh erent diversity functio ns ar e homo- geneou s polyno mials, in particular a pa cking gain d 7− → α d (resp. d 7− → α d ), α > 1 , turns out to be equ i valent to co ding with effectiv e power α 2  (resp. α 2  ). Thus, the div ersity sum , which has been kn own as a low SNR design criterio n in the literature, also scales the SNR itself, an d has the refore an impact on the highe r or der terms in the diversity fun ctional, in p articular onto th e diversity pr oduct. From this insight it is reasonable to consider the co de de sign as a constraint pack ing problem . T his mean s, that the maximizatio n o f diversity can be split u p into a two-stage optim ization procedu re: 1) Find goo d pa ckings in the codin g spa ces V C n t ,T , G C n t ,T 2) Find a transfor mation which maps the packing s into equiv a lent pa ckings with maxima l d iv ersity pr oduct. Details abou t th e optimality criteria in this c ontext can be found in [ 15]. Another important point regar ding packin g g ains is the result obtained in [14, Corollary IV .2]: The achiev able minimal distances d 2 , resp. d 2 can be lower b ounde d by a quantity which gr ows propo rtionally to T n t , thus there is a b enefit for cod e designs with la rge block leng ths and the cod es constructed in th is work b enefit conside rably in perfo rmance as we will see later on. Since th e overall com plexity o f code design and d ecoding grows also with large b lock lengths, in [15, Prop. III. 4] the inequality D iv ≤ D iv has been established , which is the diversity analogue of the info rmation theoretic in equality I ( X ; Y ) ≤ I ( X ; ( Y , H )) . From this one infers immediately that any non -cohere nt code can be u sed in a cohere nt scena rio without p erforma nce loss. Moreover [15, Th m. III .5] states, that, given a no n-coh erent code C , the set { Φ u | Φ ∈ C , u ∈ ¯ C } for any n t -by- n t coheren t co de ¯ C is actually a c oherent space time co de with diversity as least as good as the div er sities of C and ¯ C . Th is result can be interpreted as a comp lexity r eduction , providing two le vel co de design and decod ing algorithms. I V . S PAC E T I M E PAC K I N G S F RO M S P H E R I C A L C O D E S Let us star t with th e p roposed first stage optimizatio n proced ure for cod e design, namely th e con struction of p ack- ings in V C n t ,T resp. G C n t ,T with large minimal distance. A compreh ensive standard sour ce o n the g eneral sphere p acking problem in Euc lidean space is [20]. U nfortun ately the methods in [20] rely o n th e symmetr y gro up o f Euclidean space and do not app ly to our situation, where the co ding spaces are non -flat and th e distance metric is no nlinear . Although [21] c onsiders Grassmannian pac kings, it applies to th e real Grassmannian manifold only . Some genuine complex Gr assmannian packings have been constructed nu merically in [2 2],[23], and [24] but numerical o ptimization techn iques are co mputation al co mplex and give o nly little insigh t into the constru ction mech anisms nor do they possess any algeb raic stru cture. Therefo re it w ould be de sirable to find simple model spaces, where structur ed pack ings can be construc ted and then trans- formed into packing s on th e co mplex Stiefel and Gra ssmann manifold s. On the one h and this m odel space must possess a large sym metry gro up such that som e structu red packing algorithm may be d ev e loped. On the other hand it mu st be ’ similar’ to th e Stiefel a nd Grassman n manifold in order to construct a mapp ing which app roximate ly preserves (minimal) distances. In this pa per such a model space with corre sponding mapping will be presented utilizing the homogeneo us structure of the coding spaces ( compare [2 5] for a general introd uction to hom ogeneo us spaces or [26] fo r the homog eneous stru cture of th e (real) Stiefel a nd G rassmann man ifolds). I n p articular the (c omplex) Stiefel manifo ld V C n t ,T is d iffeomorphic to a coset space with respect to the unitary grou p U ( T ) of T -by - T unitary matrices: V C n t ,T ∼ = U ( T )  1 0 0 U ( T − n t )  (16) whereas ∼ = means ’ d iffeomorphic to’. This fact is du e to th e symmetry action Φ 7− →  1 0 0 U ( T − n t )  Φ leaving ( 1 0 ) fixed. Similarly for the (complex) Grassmann manifold G C n t ,T of n t dimensiona l subspaces h Φ i of C T : Since Φ 7− → h Φ i is a projectio n in variant under a ll n t -by- n t unitary basis transform ations we ob tain the coset rep resentation G C n t ,T ∼ = U ( T ) . U ( n t ) 0 0 U ( T − n t )  (17) Homogen eity (o r co set structure) mean s, tha t any two po ints can be mapped isom etrically into e ach other, in particular all distance relatio ns ar e u niquely determin ed with respect to a n arbitrarily ch osen ref erence poin t (e.g. ( 1 0 ) , re sp. h ( 1 0 ) i ). W e will see that homo geneity provid es the required ’ similarity’ mentioned above. Let us de fine D by D = dim R V C n t ,T = n t (2 T − n t ) resp. D = dim R G C n t ,T = 2 n t ( T − n t ) . The D dimensiona l sphere S D := { x ∈ R D +1 | k x k = 1 } ⊂ R D +1 is also hom ogeneo us, since it has the coset repr esentation S D = V R 1 ,D +1 ∼ = O ( D + 1)  1 0 0 O ( D )  (18) where O ( D ) denote s th e set of D -by- D orth ogonal matrices. The sphere is h ighly symmetric and ’ similar’ to our coding spaces, since in [ 14] a relation between p acking densities of th e coding spaces and S D has been establishe d, an d in [16], [1 7] a correspond ing m apping o f packin gs S D − → V C n t ,T , resp. S D − → G C n t ,T a) has been define d, u tilizing th e homog eneous coset structure. Due to th e analysis in [14] th is mapping is distance pr eserving up to a positive scaling factor . In summary , spher ical codes can be transform ed into space time codes with con trolled d istance loss. Mor eover th e theory of spherical pack ings (i.e. packing s of spher ical caps on S D ) is alre ady an item o f curren t research, see e.g . [27], [28]. Nev ertheless, here anoth er spherical packin g algor ithm will be presented to obtain structured and at the same time full rate spherical packing s. Howev e r , in the space freque ncy co ntext of MIMO- OFDM system s spherica l packing s based on lattice construction s have alre ady been in vestigated [16], [17]. a) Actuall y the mapping is appropria tely defined on the upper (or lo wer) hemisphere of S D only . This is due to the projecti ve nature of G C n t ,T such that antipod al points on the s phere will be ident ified under this mapping. V . S P H E R I C A L PAC K I N G S F RO M P E R M U TA T I O N C O D E S A mor e flexible algeb raic tool than lattices to prod uce spherical p ackings are gro ups, i.e. finite subgroups of the orthog onal group . The idea behind it is to take som e initial ( D + 1 ) dimension al vector o f un it norm (s.t. it can be considered as a poin t on the D d imensional sphere S D ). The n let th e finite subgrou p G act on the initial vector x and the outcome is a spher ical packing whose constellation size equals the o rder o f G . Th e optimizatio n procedu re to m aximize the packing d istance inv olves the choice o f the g roup G itself and the choice of the initial vector . The packings generated by such a pro cedure are ca lled geometr ically unifo rm and have been considered r ecently in a frame th eoretic co ntext [29] (see [30] for an in troductio n to frame th eory in comm unication s). In a broade r context the set of vectors (inpu t sequences) obtained as o rbits of (a sub set of) G of some initial vector is called a g roup code for the Gaussian cha nnel. This class of cod es com prises many sign al sets that ar e used in p ractice, e.g. linear binary codes. I n the special case G consisting of ( D + 1) -by - ( D + 1) matrix representation s o f permu tations, the resulting g roup code is called permutation modulation [31]. Note that in p ractice only subgro ups of the pe rmutation group will be of in terest, otherwise th e huge number of D ! permutatio ns g enerate per mutation modulation s no practical device can h andle. The correspo nding spher ical packings will be the starting point for th e following analy sis. In [31] an optimizatio n proced ure similar to a Lag rangian m ethod is presen ted, which solves fo r the initial vector wh ose gen erated per mutation modulatio n h as largest minimal distance un der the action of a fixed permutatio n subgr oup. The size o f the subgroup is specified in ter ms of the initial vector with appropriate repetitions of its compo nents x = ( µ ( m 1 ) 1 , . . . , µ ( m k ) k ) (19) where µ ( m i ) i denotes µ i repeated m i times. Alth ough the analysis in [3 1] d oes not pr ovide a complete solu tion ( no solution for the ’Lag rangian’ parameters has been given), the method reveals some stru cture o f the optimal in itial vector: The entries µ i are symmetr ically arran ged aroun d zero and the correspo nding weigh ts m i = ⌊ e − ( η + µ 2 i ) /λ ⌉ are determined accordin g to som e discrete Gaussian distribution inv olving th e ’Lagran gian’ p arameters ( η , λ ) [31, Sec. IV]. Plugging this into the constrain t equation of the ’Lagra ngian’ a nalysis yields, using M aple, complete solution s. Unfo rtunately due to th e integer constrain t on the m i solutions are possible on ly for carefully selected parameters. The typical spherical dim en- sions D occurring he re do not per mit solutions with small enoug h rates. Th erefore anoth er strategy has been c hosen. Inspection of the initial solution v ectors wit h lowest po ssible rate, such that the ’Lag rangian ’ functional provides a solu tion, revealed that there are only a few possible altern ativ es for the choice of x , namely x is ch aracterized by a large amou nt of zero com ponents and o nly a fe w non -zero o nes. The more distinct co mponen ts in x , the larger the set of distinct pe rmu- tations (high ra te), and the smaller the final minimal distance. Therefo re for pr escribed dim ension and rate the initial vector x with largest p ossible number o f zero- compo nents has been chosen, such th at th e rate requ irement is satisfied. Having fou nd a n appro priate initial vector the p roblem o f carefully selecting th e correspon ding perm utations remains. Giv e n x ∈ R D +1 of the fo rm (19) the corr espondin g numb er of distinct pe rmuted versions is (in m ulti index notation with respect to the vector m = ( m 1 , . . . , m k ) ) M :=  | m | m !  = ( P i m i )! m 1 ! . . . m k ! (20) Giv e n a prescribed space time code rate R , the corresp onding rate o f the sph erical code is r := T D +1 R a nd the required number of perm utations is given as N = ⌈ 2 ( D +1) r ⌉ , where we hav e chosen the initial vector x (resp. the vector m ) su ch that N ≤ M hold s. Then the task is, to select N o ut of the M distinct permutation s o f the multiset b) x such that the re sulting p acking has large m inimal d istance. T aking the number of transpositions require d to transform a permu tation p into ano ther perm utation q as a d istance measure between p an d q , the objective is to select N out o f M multiset permutatio ns with large pairwise d istance. I n co ntrast to or- dinary p ermutation s the structu re of multiset perm utations is more co mplicated, and there seems to be no ranking algo rithm av ailable. Ne vertheless all multiset per mutations can b e listed in Gray code o rder, wh ich is the ap propria te order ing with respect to the permu tation distance just define d. The a lgorithm can be obtain ed as a sho rt C progr am from the Com binatorial Object Server c) . Then, taking each ⌊ M N ⌋ ’ s multiset permu tation produ ced b y this algorith m does the job and we end up w ith the desired spherical pack ing with large minimal distanc e, correspo nding to th e specified rate. V I . F U L L D I V E R S I T Y RO TA T I O N Let us now come the the second stage of diversity optimiza- tion in the sense described in III-C, namely to define a distance preserving mappin g which transfor ms the space time p ackings into an equivalent packing with maximu m diversity produ ct. T o this end we precod e the space time code symbo ls by per- forming a rotation on the s pherical code as follows. As th e axis of ro tation we choose the ’ diag onal’ e = (1 , . . . , 1 ) ∈ R D +1 . Define a u nitary ( D + 1) -by- ( D + 1) matrix W e by prescribing its first row to be e/ √ D + 1 and for j = 2 , . . . , D + 1 its j th row to be (1 ( j − 1) , − j ( j − 1) , 0 ( D +1 − j ) ) / p j ( j − 1) . Clearly e = e 1 W e holds with e 1 = (1 , 0 , . . . , 0 ) , thus e 1 = eW t e , where the superscript t denotes transposition . Supp ose we already ha d defin ed a ro tation matrix R 1 with e 1 as its axis, the n we o btain the same rotation ab out the axis e as R := W t e R 1 W e . The rotation R 1 is constructed easily: Set 0 = (0 ( D ) ) , then R 1 =  1 0 0 t exp( αX )  perfor ms a rotation ab out α degrees a bout the axis e 1 , whe re X being the antisymme tric b) the term multiset denotes a set with repea ted elements c) Programmer: Frank Ruske y / Joe Sawa da http://www.the ory.csc.uvic.ca/ ˜ cos/inf/mult/M ultiset.html D × D matrix with on es on its upp er triang ular part ( which unifor mly weig hts the available degrees of fr eedom) . Figur e 1 demon strates the effect of rotation fo r som e values of α on the perfor mance of a samp le non-co herent 8 × 2 cod e of ra te 1 / 2 . No te tha t without rotation ( α = 0 , thick d ashed line ) the code d oes no t achieve full diversity o rder . T rying some values for α rev ea ls some o scillatory behavior of the codin g gain ( i.e. the value of the diversity). It turns out tha t for non-c oherent codes α = 7 4 π is a go od choice, while fo r coher ent codes α = π yields goo d results. If a non-c oherent code will be used in th e coheren t scenario by compo sing it with some small coheren t code (co mpare III- C), the angle α = 7 4 π remains a good choice. Fig. 1. Performance of R = . 5 , 8 × 2 space time codes coming from the same spherica l code, but precoded with diffe rent rotation angles V I I . S I M U L AT I O N R E S U LT S All simu lations have b een perf ormed in a scenario with n t = 2 tran smit anten nas and n r = 1 receive an tennas with maximum likelihoo d deco ding. Figure 2 d isplays the bit e rror Fig. 2. Non-cohere nt performanc e gain with increasing block length, compared to the optimal 2-by-2 dif ferential code perfor mance of a series of two-stage-o ptimized non -coher ent codes with rate app roximately one and block length s varying from 4 to 12 (continuou s line s). The corresp onding initial vectors (of dimension D + 1 ) and the number of chosen multiset perm utations are x = (0 (7) , 1 (2) ) / √ 2 , N = 32 ; x = ( − 1 , 0 (23) , 1) / √ 2 , N = 5 12 ; x = (0 (38) , 1 (3) ) / √ 3 , N = 8 1 92 , respectively . The r otation angle is α = 7 4 π an d the final space time code is th en given as th e image of the map S D − → G C n t ,T (compar e section I V), where now (for n t = 2 fix) D = 8 , 24 , 4 0 for T = 4 , 8 , 12 respectively . Note that the cardin ality o f the final space time cod es differs from the correspo nding spherical code card inality d ue to the restriction to o ne hem isphere o f S D , co mpare footno te a) in section IV (e.g. th e sph erical code o f car dinality N = 32 shr unk to a space time cod e of cardin ality 21 only , thus R ≈ 1 . 1 ) . The simulation shows th at the bit error perfor mance inc reases with the block length in perfect co nformity with the result of earlier work [1 4], mentio ned in III-C. Moreover [12] pr esented a non-co herent 2 -by-2 differential code with op timal diversity sum and div er sity produ ct. Th e perform ance of this optimal 2-by- 2 code is also shown in fig. 2 (thick d ashed line) . The compariso n re veals that the ad ditional degrees of freed om provided by the larger block lengths of the new codes based on permutatio n co des result in an approximately 2 dB performance gain over the 2-by- 2 differential code [1 2]. No te th at the non- coheren t codes constructed here a re not based o n a differential transmission scheme. Thu s the achieved performan ce gain over one of the b est known differential sch emes justifies the research effort f or non- differential sche mes. Fig. 3. Coherent performance gain with increa sing block length, compared to the well kno wn BPSK Alamouti scheme Figure 3 displays the bit error p erform ance o f a series of two-stage-optimized comp osed coheren t codes with rates ranging from 1 .64 to 0 .79 and block len gths T = 4 , 8 , 16 (continu ous lin es). They have b een com posed from a series of non-co herent codes and a QPSK Alamo uti scheme [3]. The non-co herent cod es co me from co rrespond ing spherica l co des of size N = 8 , 32 , 512 (where again some spherical code points have been removed du e to the restriction to o nly o ne hemispher e) and dimension D = 8 , 24 , 56 . Again the bit error perfor mance incr eases with the block leng th and co mparing the rate 1.05 8-by-2 code with the 2-by-2 B PSK Alamouti code (thick dashed gra y line in fig. 3 ) shows a perfo rmance gain of app roximately 2 dB. O f course the n ew codes suffer f rom a considerab le hig her deco ding complexity comp ared with th e Alamouti scheme, thus there is a tradeoff b etween performance and signal pro cessing. A m ore fair compariso n incorpo rating some additional signa l pr ocessing may be repre sented by th e thick dashed black line in fig . 3. I t shows the perfor mance of a 2-by- 2 code with optimal d iv ersity sum an d diversity prod uct, which is in fact identical to the o ptimal non-co herent 2-by -2 differential code [1 2] d) . T his co de perfor ms about 1 dB better than the Alamou ti scheme but compared w ith the new codes we still obtain a perfor mance gain of appr oximately 1 dB of the new 8 -by-2 code over the optimal 2-b y-2 co de. V I I I . C O N C L U S I O N S A N D F U T U R E W O R K A n ew class of space time cod es based o n sphe rical perm u- tation co des has been p resented. It has been demonstrated that the addition al d egrees of free dom p rovided by larger block lengths h elp to ach iev e better perform ance and even b eat th e bit error performan ce of 2 -by-2 di versity- optimal schemes. The presented constru ction app lies both to coherent and non- coheren t co de design with a two-stage optimization pro cess which re duces the design comp lexity b y geo metrical insights affording algeb raic structures. The inheren t design complexity of coheren t codes with large block lengths can be f urther com pensated in part by reduc tion to the design of n on-coh erent codes, supplem ented by small coheren t codes. The non- coherent code de sign in turn is no t based on any differential scheme but on the pack ing theory of the Grassman n ma nifold. Howe ver, the presented co nstruction sch eme, in particu lar the u se of p ermutation cod es will be in vestigated furthe r , in order to o btain low complex decoding algorithms in the futur e. 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