Codeword-Independent Performance of Nonbinary Linear Codes Under Linear-Programming and Sum-Product Decoding

Code word-I ndependent Performance of Nonb inary Linear Codes Under Linear -Programming and Sum-Product Decoding Mark F . Flanagan DEIS/CNIT University of Bologna via V enezia 52, 4702 3 Cesen a (FC), Italy Email: mark.flanag an@ieee.org Abstract — A coded modu lation system is considered in which nonbinary coded symbols ar e mapped direc tly to nonbinary modulation signals. It is prov ed that if the modulator -chan nel combination satisfies a particular symmetry condi tion, the code- word error rate p erf ormance is independent of the transmitted codeword. It is shown that this r esult holds f or both linear - program ming decoders and sum-product decoders. In particular , this provides a natural modulation mappin g for nonbinary codes mapped to PSK constellations f or transmission ov er memoryless channels such as A WGN chan nels or flat fading channels with A WGN. I . I N T RO D U C T I O N Low-density parity check (LDPC) c odes [1] , as well as their nonbin ary counterparts [2] hav e bee n shown to ex- hibit excellent erro r-correcting perf ormance when decod ed by the traditional sum-pr od uct (SP) decoding algorithm. In [3], Feldman et al. introduced th e idea of linear -p r ogramming (LP) d ecoding of LDPC code s. This was later generalized to nonbin ary cod es in [4]. For classical coded m odulation systems, geometric unifor- mity [5] was identified as a symmetry condition which, if satis- fied, guaran tees co dew ord error rate perf ormance in depend ent of the transmitted codeword, where maximum-likelihood (ML) decodin g is assume d. Some r ecent coded m odulation schemes with SP decoding used this symmetry co ndition for design [6]. An anala gous symmetry condition was defined in [7] fo r binary cod es over GF (2) with SP deco ding; this was extended to nonbin ary codes over GF ( q ) by in voking the concept of coset LDPC co des [2]. In this work it is shown that for the cases of LP and SP decodin g of linear codes over rings, there exists a symmetry condition und er which the cod e word error rate perf ormance is indepen dent of th e transmitted cod ew o rd (fo r the case of LP decoding th is theo rem g eneralizes [3 , Theorem 6 ], and is stated in [4]). This provides a con dition som ewhat akin to geometric unifor mity fo r state-of-the-ar t non binary coded modulatio n systems. I I . G E N E R A L F R A M E W O R K W e consider codes over finite rings (this in cludes codes over fin ite fields, but may be more gen eral). Deno te by R a rin g with q elements, b y 0 its additive identity , and let R − = R \ { 0 } . Let C = { c ∈ R n : c H T = 0 } be a linear code defined with respect to the m × n parity-chec k matrix H over R . Denote the set of co lumn ind ices and the set of row indices of H b y I = { 1 , 2 , · · · , n } and J = { 1 , 2 , · · · , m } , respectively . For j ∈ J , let H ( r ) j denote the j -th row o f H , an d for i ∈ I , let H ( c ) i denote the i -th co lumn. Denote by supp ( c ) th e suppo rt of a vector c . For eac h i ∈ I , let J i = sup p ( H ( c ) i ) and fo r each j ∈ J , let I j = sup p ( H ( r ) j ) . Also let A j,i = I j \{ i } and D j,i = J i \{ j } . Giv en any c ∈ R n , we say that parity check j ∈ J is satisfied b y c if and only if X i ∈I j c i · H j,i = 0 (1) For j ∈ J , define the single parity check code C j by C j = { ( b i ) i ∈I j : X i ∈I j b i · H j,i = 0 } Note that while th e symbols o f the cod e words in C are ind exed by I , the symbols o f the codew ords in C j are indexed by I j . W e define the projection mappin g for par ity check j ∈ J by x j ( c ) = ( c i ) i ∈I j Then, gi ven any c ∈ R n , we m ay say that par ity check j ∈ J is satisfied by c if and on ly if x j ( c ) ∈ C j (2) since (1) an d (2) are eq uiv alen t. Also, we say that the vector c is a codeword of C , writing c ∈ C , if and only if all parity checks j ∈ J are satisfied by c . Assume th at the codeword ¯ c = ( ¯ c 1 , ¯ c 2 , · · · , ¯ c n ) ∈ C h as been tran smitted over a q -ary in put memo ryless channel, and a corrup ted word y = ( y 1 , y 2 , · · · , y n ) ∈ Σ n has been receiv ed. Here Σ den otes the set o f cha nnel o utput symb ols; we assum e that this set either has finite card inality , or is eq ual to R l or C l for so me integer l ≥ 1 . In practice, th is cha nnel ma y represent the combination of mod ulator and physical chan nel. It is assumed hereafter th at all information words a re equally probab le, and so all codew o rds are tr ansmitted with equal probab ility . Next we set up some definitions and notatio n. W e define the mapp ing ξ : R 7→ { 0 , 1 } q − 1 ⊂ R q − 1 by ξ ( α ) = x = ( x ( γ ) ) γ ∈ R − such that, for each γ ∈ R − , x ( γ ) =  1 if γ = α 0 otherwise. W e note that th e m apping ξ is o ne-to-on e, and its image is the set of bin ary vectors o f len gth q − 1 with Hamming weight 0 or 1. Building on this, we also d efine Ξ : R n 7→ { 0 , 1 } ( q − 1) n ⊂ R ( q − 1) n accordin g to Ξ ( c ) = ( ξ ( c 1 ) | ξ ( c 2 ) | · · · | ξ ( c n )) W e note that Ξ is also one-to -one. Now , for vectors f ∈ R ( q − 1) n , we adopt the notation f = ( f 1 | f 2 | · · · | f n ) where ∀ i ∈ I , f i = ( f ( α ) i ) α ∈ R − In particular, we define λ ∈ R ( q − 1) n by setting, for each i ∈ I , α ∈ R − , λ ( α ) i = log  p ( y i | 0) p ( y i | α )  and p ( y i | c i ) deno tes th e c hannel output probab ility (de nsity) condition ed on the ch annel input. Also, we ma y use this no tation to write the inverse of Ξ as Ξ − 1 ( f ) = ( ξ − 1 ( f 1 ) , ξ − 1 ( f 2 ) , · · · , ξ − 1 ( f n )) I I I . D E C O D I N G A L G O R I T H M S A. Linear-Pr ogramming Decoder The linear-programmin g (LP) decoder of [4] oper ates as follows. The lin ear p rogram descr ibed her e is equivalent to that gi ven in [4 ]; ho wev er , so me chan ges o f no tation have been mad e in order to facilitate the proof to co me in section IV. The variables o f the LP are f ( α ) i for each i ∈ I , α ∈ R − and w j, b for each j ∈ J , b ∈ C j and the constraints are ∀ j ∈ J , ∀ b ∈ C j , w j, b ≥ 0 (3) and ∀ j ∈ J , X b ∈ C j w j, b = 1 (4) and ∀ j ∈ J , ∀ i ∈ I j , ∀ α ∈ R − , f ( α ) i = P b ∈ C j , b i = α w j, b (5) The set of p oints ( f , w ) which satisfy (3)-(5) form a polytope denoted by Q . The cost function to be minimized over this polytop e is F ( f ) = λf T , and the m inimizer is den oted b y ˆ f . If ˆ f ∈ { 0 , 1 } ( q − 1) n , th e ou tput is the codeword Ξ − 1 ( ˆ f ) (it is proved in [4] th at this must be the maximu m-likelihood co de- word). Otherwise, the decoder outputs a ‘decod ing failure’ . B. Sum-Pr o duct Deco der The sum-p roduct ( SP) decod er o perates as fo llows . Note that in p ractice, compu tations a re usually carr ied out in the log-do main, but this does not affect ou r analysis. Initializing m i ( α ) = p ( y i | α ) ∀ i ∈ I , ∀ α ∈ R (6) and m D, 0 j,i ( α ) = 1 ∀ j ∈ J , ∀ i ∈ I j , ∀ α ∈ R (7) N iterations of fully parallel SP decodin g may b e represented by the fo llowing recu rsiv e form ulas. F or each k = 1 , 2 , · · · N , m U,k j,i ( α ) = m i ( α ) · Y l ∈D j,i m D,k − 1 l,i ( α ) (8) for each j ∈ J , i ∈ I j , α ∈ R , and m D,k j,i ( α ) = X P l ∈A j,i d l H j,l = − α H j,i    Y l ∈A j,i m U,k j,l ( d l )    (9) for each j ∈ J , i ∈ I j , α ∈ R . Finally , decisions are made via g i ( α ) = m i ( α ) · Y j ∈J i m D,N j,i ( α ) ∀ i ∈ I , ∀ α ∈ R (10) and h i = arg max α ∈ R { g i ( α ) } ∀ i ∈ I (11) The output of the decod er is th en h = ( h 1 , h 2 , · · · , h n ) . I V . M A I N R E S U L T Symmetry Condition. For each β ∈ R , th ere exists a bijection τ β : Σ − → Σ such that the c hannel o utput p robability (density) conditioned on the channel inpu t satisfies p ( y | α ) = p ( τ β ( y ) | α − β ) (12) for all y ∈ Σ , α ∈ R . When Σ is equal to R l or C l for l ≥ 1 , the m apping τ β is assumed to be isometric with respect to Euclidean distance in Σ , for every β ∈ R . In the f ollowing, cod ewor d err or is defined as the ev ent where the decoder output is not equal to the transmitted codeword. 2 Theor em 4.1: Under the stated symmetry condition, the probab ility of codeword error is indepen dent of the transmitted codeword (a) under linear-program ming decoding (b) under sum-pr oduct d ecoding . Pr oof: W e shall prove th e theorem for the case wher e Σ has infinite cardinality; the c ase of discrete Σ may be ha ndled similarly . Fix some codeword c ∈ C , c 6 = 0 . W e wish to prove that Pr ( Err | c ) = Pr ( Err | 0 ) where Pr ( Err | c ) denotes the probab ility of co dew o rd error giv en that the codew ord c was transmitted. Now Pr ( Err | c ) = Pr ( y ∈ B ( c ) | c ) where B ( c ) is the set of all receiv e words which may cause codeword erro r , gi ven that c was tr ansmitted. Also Pr ( Err | 0 ) = Pr ( y ∈ B ( 0 ) | 0 ) So we write Pr ( Err | c ) = Z y ∈ B ( c ) p ( y | c ) d y (13) and Pr ( Err | 0 ) = Z ˜ y ∈ B ( 0 ) p ( ˜ y | 0 ) d ˜ y (14) Now , setting α = β in the symmetry cond ition (12) y ields p ( y | β ) = p ( τ β ( y ) | 0) (15) for any y ∈ Σ , β ∈ R . W e now de fine ˜ y = G ( y ) as follows. For ev ery i ∈ I , if c i = β ∈ R then ˜ y i = τ β ( y i ) W e note that G is a bijection fro m the set Σ n to itself, and that if y , z ∈ Σ n and c i = β ∈ R then k y i − z i k 2 = k τ β ( y i ) − τ β ( z i ) k 2 and so k G ( y ) − G ( z ) k 2 = k y − z k 2 i.e. G is isometric with respect to Eu clidean distance in Σ n . W e prove that the integral (13) may be transformed to (14) via the substitution ˜ y = G ( y ) . Fir st, we have p ( y | c ) = Y i ∈I p ( y i | c i ) = Y β ∈ R Y i ∈I , c i = β p ( y i | β ) = Y β ∈ R Y i ∈I , c i = β p ( τ β ( y i ) | 0) = Y β ∈ R Y i ∈I , c i = β p ( ˜ y i | 0) = Y i ∈I p ( ˜ y i | 0) = p ( ˜ y | 0 ) Since G is isom etric with r espect to E uclidean d istance in Σ n , it follows that the Jacobian d eterminant of the transfo rmation is equ al to un ity . Ther efore, to comple te the pro of, we need only show th at y ∈ B ( c ) if and on ly if ˜ y ∈ B ( 0 ) W e prove this sep arately f or th e two cases of linear- progr amming and sum- produ ct decoding. (a) Under linear-pr ogramming decoding: Here B ( c ) = { y ∈ Σ n : ∃ ( f , w ) ∈ Q , f 6 = Ξ ( c ) with λf T ≤ λ Ξ ( c ) T } Recall that here λ is a fu nction o f y via λ ( α ) i = log  p ( y i | 0) p ( y i | α )  (16) for i ∈ I , α ∈ R − . Also B ( 0 ) = { ˜ y ∈ Σ n : ∃ ( ˜ f , ˜ w ) ∈ Q , ˜ f 6 = Ξ ( 0 ) with ˜ λ ˜ f T ≤ ˜ λ Ξ ( 0 ) T } Here ˜ λ is a functio n o f ˜ y via ˜ λ ( α ) i = log  p ( ˜ y i | 0) p ( ˜ y i | α )  (17) for i ∈ I , α ∈ R − . W e b egin by r elating th e elements of λ (defined by (16)) to the elements of ˜ λ (d efined by (1 7)). Let i ∈ I , α ∈ R − . Suppose c i = β ∈ R . W e then have λ ( α ) i = log  p ( y i | 0) p ( y i | α )  = log  p ( τ β ( y i ) | − β ) p ( τ β ( y i ) | α − β )  = log  p ( ˜ y i | − β ) p ( ˜ y i | α − β )  This yields λ ( α ) i =      ˜ λ ( α ) i if β = 0 − ˜ λ ( − α ) i if α = β ˜ λ ( α − β ) i − ˜ λ ( − β ) i otherwise. Next, for any point ( f , w ) ∈ Q we define a new point ( ˜ f , ˜ w ) as follows . For all i ∈ I , α ∈ R − , if c i = β ∈ R then ˜ f ( α ) i = ( 1 − P γ ∈ R − f ( γ ) i if α = − β f ( α + β ) i otherwise. (18) For all j ∈ J , r ∈ C j we define ˜ w j, r = w j, b where b = r + x j ( c ) Next we prove that for e very ( f , w ) ∈ Q , the new poin t ( ˜ f , ˜ w ) lies in Q and thus is a feasible solu tion for the LP . Constraints ( 3) and (4) obviou sly hold f rom the definition of 3 ˜ w . T o verify (5), we let j ∈ J , i ∈ I j and α ∈ R − . W e also let c i = β ∈ R . W e now check two cases: • If α = − β , ˜ f ( α ) i = 1 − X γ ∈ R − f ( γ ) i = X b ∈C j w j, b − X γ ∈ R − X b ∈C j , b i = γ w j, b = X b ∈C j , b i =0 w j, b = X r ∈ C j , r i = α ˜ w j, r • If α 6 = − β , ˜ f ( α ) i = f ( α + β ) i = X b ∈C j , b i = α + β w j, b = X r ∈ C j , r i = α ˜ w j, r Therefo re ( ˜ f , ˜ w ) ∈ Q , i.e. ( ˜ f , ˜ w ) is a feasible solution for the LP . W e write ( ˜ f , ˜ w ) = L ( f , w ) . W e also note that the mapping L is a bijectio n fr om Q to itself; this is easily shown by verifying the in verse f ( α ) i = ( 1 − P γ ∈ R − ˜ f ( γ ) i if α = β ˜ f ( α − β ) i otherwise (19) for all i ∈ I , α ∈ R − , and w j, b = ˜ w j, r where r = b − x j ( c ) for all j ∈ J , b ∈ C j . W e now prove that fo r ev ery ( f , w ) ∈ Q , ( ˜ f , ˜ w ) = L ( f , w ) satisfies λf T − λ Ξ ( c ) T = ˜ λ ˜ f T − ˜ λ Ξ ( 0 ) T (20) W e achieve this by proving λ i f T i − λ i ξ ( c i ) T = ˜ λ i ˜ f T i − ˜ λ i ξ (0) T (21) for every i ∈ I . W e may then obtain (20) by summ ing (21) over i ∈ I . Let c i = β ∈ R . W e conside r two cases: • If β = 0 , (21) becom es λ i f T i = ˜ λ i ˜ f T i which holds since in this c ase ˜ λ ( α ) i = λ ( α ) i and ˜ f ( α ) i = f ( α ) i for all α ∈ R − . • If β 6 = 0 , λ i f T i − λ i ξ ( c i ) T = X γ ∈ R − λ ( γ ) i f ( γ ) i − λ ( β ) i = X γ ∈ R − γ 6 = β  ˜ λ ( γ − β ) i − ˜ λ ( − β ) i  f ( γ ) i − ˜ λ ( − β ) i f ( β ) i + ˜ λ ( − β ) i = X α ∈ R − α 6 = − β ˜ λ ( α ) i f ( α + β ) i + ˜ λ ( − β ) i   1 − X γ ∈ R − f ( γ ) i   = X α ∈ R − ˜ λ ( α ) i ˜ f ( α ) i = ˜ λ i ˜ f T i − ˜ λ i ξ (0 ) T where we ha ve made use of the sub stitution α = γ − β in the third line. Therefo re (21) h olds, proving (20). Finally , we note that it is easy to sho w , using (18) a nd (19), that f = Ξ ( c ) if an d only if ˜ f = Ξ ( 0 ) . Putting together these results, we may make the following statement. Supp ose we ar e given y , ˜ y ∈ Σ n with ˜ y = G ( y ) . Then the point ( f , w ) ∈ Q satisfies f 6 = Ξ ( c ) and λf T ≤ λ Ξ ( c ) T if and o nly if the point ( ˜ f , ˜ w ) = L ( f , w ) ∈ Q satisfies ˜ f 6 = Ξ ( 0 ) a nd ˜ λ ˜ f T ≤ ˜ λ Ξ ( 0 ) T . This statement, along with the fact that both G and L are bijective, proves that y ∈ B ( c ) if and on ly if ˜ y ∈ B ( 0 ) This completes the proof of the theo rem f or the case of LP decodin g. (b) Under Sum-Pr oduct De coding: Recall th at all decod er variables ap pearing in equatio ns (6 )- (11) are function s of y via (6). For any such v ar iable x , let ˜ x denote the correspond ing variable with ˜ y as in put. Then we have, for all i ∈ I , α ∈ R , where c i = β , m i ( α ) = p ( y i | α ) = p ( τ β ( y i ) | α − β ) = p ( ˜ y i | α − c i ) = ˜ m i ( α − c i ) Next we prove by indu ction that f or all k = 0 , 1 , · · · N , m D,k j,i ( α ) = ˜ m D,k j,i ( α − c i ) (22) for all j ∈ J , i ∈ I j , α ∈ R . Th is r esult holds f or th e base case k = 0 b ecause from (7) m D, 0 j,i ( α ) = ˜ m D, 0 j,i ( α ) = 1 ∀ j ∈ J , ∀ i ∈ I j , ∀ α ∈ R Assuming that (22) holds for some k = r − 1 ∈ { 0 , 1 , · · · N − 1 } (and for all j ∈ J , i ∈ I j , α ∈ R ), we obtain by (8) m U,r j,i ( α ) = m i ( α ) · Y l ∈D j,i m D,r − 1 l,i ( α ) = ˜ m i ( α − c i ) · Y l ∈D j,i ˜ m D,r − 1 l,i ( α − c i ) = ˜ m U,r j,i ( α − c i ) 4 for all j ∈ J , i ∈ I j , α ∈ R . So, by (9), m D,r j,i ( α ) = X P l ∈A j,i d l H j,l = − α H j,i    Y l ∈A j,i m U,r j,l ( d l )    = X P l ∈A j,i d l H j,l = − α H j,i    Y l ∈A j,i ˜ m U,r j,l ( d l − c l )    = X P l ∈A j,i b l H j,l = − ( α − c i ) H j,i    Y l ∈A j,i ˜ m U,r j,l ( b l )    = ˜ m D,r j,i ( α − c i ) for all j ∈ J , i ∈ I j , α ∈ R , where we h a ve m ade the substitution b l = d l − c l for e ach l ∈ I j , a nd used the fact that P l ∈A j,i c l H j,l = − c i H j,i since c ∈ C . It fo llows by the principle of indu ction that (22) hold s for ev ery k = 0 , 1 , · · · N , j ∈ J , i ∈ I j , α ∈ R . Therefore by (10) g i ( α ) = m i ( α ) · Y j ∈J i m D,N j,i ( α ) = ˜ m i ( α − c i ) · Y j ∈J i ˜ m D,N j,i ( α − c i ) = ˜ g i ( α − c i ) for all i ∈ I , α ∈ R , and so by (11), ˜ h i = h i − c i for all i ∈ I . The refore h 6 = c if an d only if ˜ h 6 = 0 . W e co nclude that y ∈ B ( c ) if and only if ˜ y ∈ B ( 0 ) This completes th e p roof of the theorem for the case of SP decodin g. I t is trivial to see that this proof gener alizes to the case of optio nal early exit of the iterati ve loop on successful completion of a syndr ome ch eck. V . A P P L I C A T I O N : N O N B I N A RY C O D E S M A P P E D T O P S K M O D U L AT I O N While th is theo rem may be shown to app ly to other coded modulation s ystems such as nonbinary coded orthogonal modulatio n over memoryless channels and nonbinar y codin g over the discrete m emoryless q -ary symmetric channel, we focus in this paper on th e practical applicatio n of nonbin ary codes mapped directly to PSK symb ols a nd transmitted over a memor yless channel. Here Σ = C , and deno ting the ring elements by R = { a 0 , a 1 , · · · , a q − 1 } , the modu lation mapping may be written withou t loss of generality as M : R 7→ C such that M ( a k ) = exp  ı 2 π k q  (23) for k = 0 , 1 , · · · , q − 1 (here ı = √ − 1 ). Here (15), to gether with the r otational symm etry o f the q -ar y PSK con stellation, motiv ates us to define, fo r e very β = a k ∈ R , τ β ( x ) = exp  − ı 2 π k q  · x ∀ x ∈ C (24) Next, we also impo se the con dition that R und er addition is a cyclic group. T o see why we impose this co ndition, let α = a k ∈ R and β = a l ∈ R . By th e symmetry c ondition we must hav e p ( y i | α + β ) = p ( τ α + β ( y i ) | 0) and also p ( y i | α + β ) = p ( τ β ( y i ) | α ) = p ( τ α ( τ β ( y i )) | 0) In order to e quate these two expr essions, we impose the condition τ α + β ( x ) = τ α ( τ β ( x )) for all x ∈ C , α, β ∈ R . Letting α + β = a p ∈ R , and u sing (24) yields exp  − ı 2 π k q  · exp  − ı 2 π l q  = exp  − ı 2 π p q  and thus p ≡ k + l mod q . Therefo re, we mu st h av e a k + a l = a ( k + l m od q ) (25) for all a k , a l ∈ R . This im plies that R , under additio n, is a cyclic group. It is easy to check that the con dition that R un der add ition is cyclic, encapsulated by (25), along with the modulation mapping ( 23), satisfies the sy mmetry co ndition, where the approp riate ma ppings τ β are given by (24). This means that codeword-indepen dent performance is guara nteed f or such systems using nonbinar y codes with PSK m odulation . This applies to A WGN, flat fading wireless chann els, and OFDM systems transmitting over f requency selecti ve chan nels with sufficiently long cyclic prefix. A C K N OW L E D G M E N T The author would l ike to thank M. Greferath and V . Sk achek for provid ing h elpful commen ts which improved th e presen - tation of this paper . This work was suppo rted by the Claude Shannon Institute, Dublin, Ireland (Science Foundation Ireland Grant 06/MI /006), the Uni versity o f Bolo gna (ESRF-ISA) and the EC-IST Optimix project (IST -2146 25). R E F E R E N C E S [1] R. G. 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