Polytope Representations for Linear-Programming Decoding of Non-Binary Linear Codes

Polytope Representations for Li near -Programming Decoding of Non-Binary Line ar Codes V italy Skache k Claude Shannon Institute University College Dublin Belfield, Dublin 4 , Ireland vitaly .skache k@ucd.ie Mark F . Flanaga n DEIS University of Bolog na 47023 Cesena, I taly mark.flan agan@ieee.org Eimear Byrne 1 School of Mathematical Sciences University College Dublin Belfield, Du blin 4, Ir eland ebyrne @ucd.ie Marcus Greferath 1 School of Mathematical Scienc es University College Dublin Belfield, Du blin 4, Ir eland marcus.gr eferath@ucd .ie Abstract — In previous work, we demonstrated how decodin g of a non-binary linear code could be f ormulated as a li near - program ming problem. In this paper , we stud y different poly- topes for use with linear -p rogr amming decoding, and show that fo r many classes of codes these polytopes yield a complexity ad- vantage for decoding. These representations lead to polynomial- time decoders f or a wid e variety of classical non-b inary linear codes. I . I N T R O D U C T I O N In [1] and [ 2], the decoding of binary LDPC codes using linear-programm ing deco ding was prop osed, and the con- nections b etween linear-prog ramming deco ding and classical belief prop agation dec oding were established. In [3], the approa ch of [2] was e xtended to coded modulation , in p ar- ticular to codes over rings m apped to non-b inary m odulation signals. In both c ases, the principal ad vantage of the linear- progr amming framework is its math ematical tractability [2], [3]. For the binary coding framework, altern ativ e polytop e rep- resentations were stud ied which gave a co mplexity advantage in cer tain scenario s [1], [2], [ 4], [5]. Analagous to the work of [1], [2], [4], [5] f or binary codes, we defin e two p olytope representatio ns alternati ve to that prop osed i n [3] which offer a smaller numb er of variables a nd constrain ts for m any classes of n onbinar y codes. W e compare these representatio ns with the p olytope in [3]. These rep resentations are also shown to have equal error-correcting perform ance to the polyto pe in [3]. I I . L I N E A R - P RO G R A M M I N G D E C O D I N G Consider codes over finite quasi-Frob enius ring s (th is in- cludes codes over fin ite fields, b ut may be more general). Denote by R such a ring with q elements, by 0 its additiv e identity , and le t R − = R \{ 0 } . Let C be a lin ear code of length n over R with m × n p arity-check m atrix H . Denote the set of column in dices and the set of row indices of H by I = { 1 , 2 , · · · , n } and J = { 1 , 2 , · · · , m } , respectively . The no tation H j will be used f or the j -th row of H . Denote b y supp ( c ) the sup port of a vector c . For each j ∈ J , let I j = supp ( H j ) and d j = |I j | , and let d = max j ∈J { d j } . 1 These authors are also affili ated with the Claude Shannon Institute for Discrete Mathematics, Coding and Cryptography . Giv en any c ∈ R n , parity ch eck j ∈ J is satisfied by c if and only if the f ollowing equality h olds over R : X i ∈I j c i · H j,i = 0 . (1) For j ∈ J , define the single parity ch eck 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 symbo ls of the cod ew ords in C are in dexed by I , the sym bols of the codewords in C j are in dexed by I j . Observe that c ∈ C if an d o nly if all parity ch ecks j ∈ J are satisfied by c . Assume that the codeword ¯ c = (¯ c 1 , ¯ c 2 , · · · , ¯ c n ) ∈ C has been transmitted over a q -ar y in put memoryless channel, and a corrup ted word y = ( y 1 , y 2 , · · · , y n ) ∈ Σ n has been received. Here Σ d enotes the set of ch annel o utput symbols. In ad dition, assume that all c odewords are transmitted with equal probab ility . For vectors f ∈ R ( q − 1) n , the n otation f = ( f 1 | f 2 | · · · | f n ) , will be used , wh ere ∀ i ∈ I , f i = ( f ( α ) i ) α ∈ R − . W e also define a functio n λ : Σ − → ( R ∪ {± ∞} ) q − 1 by λ = ( λ ( α ) ) α ∈ R − , where, f or each y ∈ Σ , α ∈ R − , λ ( α ) ( y ) = log  p ( y | 0) p ( y | α )  , and p ( y | c ) denotes the chan nel output prob ability (density) condition ed on the chan nel input. Extend λ to a map on Σ n by λ ( y ) = ( λ ( y 1 ) | λ ( y 2 ) | . . . | λ ( y n )) . The LP d ecoder in [3] perfor ms the fo llowing co st functio n minimization : ( ˆ f , ˆ w ) = arg min ( f , w ) ∈Q λ ( y ) f T , (2) where the polytope Q is a relax ation of th e conve x hull of all points f ∈ R ( q − 1) n , wh ich corresp ond to codewords; this polytop e is defined as the set of f ∈ R ( q − 1) n , togeth er with the auxiliary variables w j, b for j ∈ J , b ∈ C j , which satisfy the fo llowing constraints: ∀ j ∈ J , ∀ b ∈ C j , w j, b ≥ 0 , (3) ∀ 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 minimization of the objective fu nction (2) o ver Q forms the relaxed L P decoding problem. The number of v ar iables and constraints for th is LP are upp er-bounded by n ( q − 1) + mq d − 1 and m ( q d − 1 + d ( q − 1) + 1 ) respectively . It is shown in [3] that if ˆ f is integral, the decoder out- put correspo nds to the max imum-likelihoo d ( ML) codeword. Otherwise, the decoder ou tputs an ‘er ror’. I I I . N E W L P D E S C R I P T I O N The results in this section are a generalization o f the h igh- density po lytope rep resentation [2, Appendix II]. Recall that the rin g R contain s q − 1 non -zero elements. Correspo ndingly , for vectors k ∈ N q − 1 , we adopt th e notation k = ( k α ) α ∈ R − Now , for any j ∈ J , we define the mapping κ j : C j − → N q − 1 , b 7→ κ j ( b ) defined b y ( κ j ( b )) α = |{ i ∈ I j : b i · H j,i = α }| for all α ∈ R − . W e may then ch aracterize the image of κ j , which we denote b y T j , as T j = ( k ∈ N q − 1 : X α ∈ R − α · k α = 0 and X α ∈ R − k α ≤ d j ) , for each j ∈ J , wh ere, for any k ∈ N , α ∈ R , α · k =  0 if k = 0 α + · · · + α if k > 0 ( k terms in sum) . The set T j is equ al to the set of all possible vectors κ j ( b ) for b ∈ C j . Note that κ j is n ot a bijection, in gen eral. W e say that a local codeword b ∈ C j is k -con strained over C j if κ j ( b ) = k . Next, for any in dex set Γ ⊆ I , we in troduc e the following definitions. L et N = | Γ | . W e define th e single-parity-ch eck- code, over vectors indexed by Γ , by C Γ = ( a = ( a i ) i ∈ Γ ∈ R N : X i ∈ Γ a i = 0 ) . (6) Also define a mapping κ Γ : C Γ − → N q − 1 by ( κ Γ ( a )) α = |{ i ∈ Γ : a i = α }| , and define, fo r k ∈ T j , C ( k ) Γ = { a ∈ C Γ : κ Γ ( a ) = k } . Below , we define a new poly tope for d ecoding . Recall that y = ( y 1 , y 2 , · · · , y n ) ∈ Σ n stands for the rec eiv ed (co rrupted ) word. In the sequel, we make u se of th e fo llowing variables: • For all i ∈ I and all α ∈ R − , we have a variable f ( α ) i . This variable is a n indicator of th e event y i = α . • For all j ∈ J and k ∈ T j , we have a variable σ j, k . Similarly to its co unterpar t in [ 2], this variable indicates the contribution to parity ch eck j o f k - constrained local codewords over C j . • For all j ∈ J , i ∈ I j , k ∈ T j , α ∈ R − , we have a variable z ( α ) i,j, k . This variable in dicates the po rtion of f ( α ) i assigned to k -co nstrained local cod ew ords over C j . Motiv ate d by these v ariable definitions, for all j ∈ J we impose the f ollowing set of constraints: ∀ i ∈ I j , ∀ α ∈ R − , f ( α ) i = X k ∈T j z ( α ) i,j, k . (7) X k ∈T j σ j, k = 1 . (8) ∀ k ∈ T j , ∀ α ∈ R − , X i ∈I j , β ∈ R − , β H j,i = α z ( β ) i,j, k = k α · σ j, k . ( 9) ∀ i ∈ I j , ∀ k ∈ T j , ∀ α ∈ R − , z ( α ) i,j, k ≥ 0 . (10) ∀ i ∈ I j , ∀ k ∈ T j , X α ∈ R − X β ∈ R − , β H j,i = α z ( β ) i,j, k ≤ σ j, k . ( 11) W e note that the fur ther con straints ∀ i ∈ I , ∀ α ∈ R − , 0 ≤ f ( α ) i ≤ 1 , (12) ∀ j ∈ J , ∀ k ∈ T j , 0 ≤ σ j, k ≤ 1 , (13) and ∀ j ∈ J , ∀ i ∈ I j , ∀ k ∈ T j , ∀ α ∈ R − , z ( α ) i,j, k ≤ σ j, k , ( 14) follow from constra ints (7)-( 11). W e d enote by U the poly tope formed by constraints (7) -(11). Let T = max j ∈J |T j | . Then , upper b ounds on the number of variables an d constrain ts in th is LP are g i ven by n ( q − 1) + m ( d ( q − 1) + 1) T and m ( d ( q − 1) + 1) + m (( d + 1)( q − 1 ) + d ) T , respectively . Since T ≤  d + q − 1 d  , th e number of variables and constraints are O ( mq · d q ) , which, for many families o f cod es, is significantly lower than the cor respond ing com plexity fo r polytop e Q . 2 For notational simplicity in proofs in this paper, it is conv enient to d efine a new set of variables as follows: ∀ j ∈ J , ∀ i ∈ I j , ∀ k ∈ T j , ∀ α ∈ R − , τ ( α ) i,j, k = X β ∈ R − , β H j,i = α z ( β ) i,j, k . ( 15) Then constraints (9 ) a nd (11) may be rewritten as ∀ j ∈ J , k ∈ T j , ∀ α ∈ R − , X i ∈I j τ ( α ) i,j, k = k α · σ j, k , (16 ) ∀ j ∈ J , ∀ i ∈ I j , ∀ k ∈ T j , 0 ≤ X α ∈ R − τ ( α ) i,j, k ≤ σ j, k . (17) Note that the variables τ do not f orm part of the LP de - scription, and therefore do n ot contribute to its complexity . Howe ver these variables will provid e a convenient notationa l shorthand f or proving results in this paper . W e will pr ove that optimizing the cost function (2) ov er this new p olytope is equiv alent to o ptimizing over Q . First, we state the f ollowing p ropo sition, wh ich will be nec essary to prove this result. Pr opo sition 3.1 : Let M ∈ N and k ∈ N q − 1 . Also let Γ ⊆ I . Assume th at for each α ∈ R − , we have a set of nonnegative integers X ( α ) = { x ( α ) i : i ∈ Γ } and that together these satisfy th e constraints X i ∈ Γ x ( α ) i = k α M (18) for all α ∈ R − , an d X α ∈ R − x ( α ) i ≤ M (19) for all i ∈ Γ . Then, th ere exist nonnegati ve integers n w a : a ∈ C ( k ) Γ o such that 1) X a ∈C ( k ) Γ w a = M . (20) 2) For all α ∈ R − , i ∈ Γ , x ( α ) i = X a ∈C ( k ) Γ , a i = α w a . (21) A sketch of the pr oof o f th is pro position will follow at the end of th is section. W e now prove the main result. Theor em 3. 2: T he set ¯ U = { f : ∃ σ , z s.t. ( f , σ , z ) ∈ U } is equal to the s et ¯ Q = { f : ∃ w s.t. ( f , w ) ∈ Q } . Therefo re, optimizing the linear cost function (2) over U is equiv alent to optimizing over Q . Pr oof: 1) Sup pose, ( f , w ) ∈ Q . For all j ∈ J , k ∈ T j , we define σ j, k = X b ∈C j , κ j ( b )= k w j, b , and for all j ∈ J , i ∈ I j , k ∈ T j , α ∈ R − , we define z ( α ) i,j, k = X b ∈C j , κ j ( b )= k , b i = α w j, b , It is straig htforward to check that constraints (10) and (11) ar e satisfied by th ese defin itions. For every j ∈ J , i ∈ I j , α ∈ R − , we have by (5) f ( α ) i = X b ∈C j , b i = α w j, b = X k ∈T j X b ∈C j , κ j ( b )= k , b i = α w j, b = X k ∈T j z ( α ) i,j, k , and thus co nstraint (7) is satisfied. Next, for every j ∈ J , we have by (4) 1 = X b ∈C j w j, b = X k ∈T j X b ∈C j , κ j ( b )= k w j, b = X k ∈T j σ j, k , and thus co nstraint (8) is satisfied. Finally , fo r every j ∈ J , k ∈ T j , α ∈ R − , X i ∈I j , β ∈ R − , β H j,i = α z ( β ) i,j, k = X i ∈I j , β ∈ R − , β H j,i = α X b ∈C j , κ j ( b )= k , b i = β w j, b = X b ∈C j , κ j ( b )= k X i ∈I j , b i H j,i = α w j, b = X b ∈C j , κ j ( b )= k k α · w j, b = k α · σ j, k . Thus, c onstraint (9) is also satisfied. This comple tes the proof of the first p art o f th e theo rem. 2) Now assume ( f , σ , z ) is a verte x of the polytop e U , and so all variables are ratio nal, a s are the variables τ . Next, fix some j ∈ J , k ∈ T j , and consider th e sets X ( α ) 0 = ( τ ( α ) i,j, k σ j, k : i ∈ I j ) . for α ∈ R − . By con straint (17), fo r each α ∈ R − , all the values in the set X ( α ) 0 are ratio nal nu mbers b etween 0 and 1. Let µ be the lowest common d enominato r of all the nu mbers in all the sets X ( α ) 0 , α ∈ R − . Let X ( α ) = ( µ · τ ( α ) i,j, k σ j, k : i ∈ I j ) , for each α ∈ R − . Th e sets X ( α ) consist o f integers between 0 and µ . By c onstraint (16), we must have that for every α ∈ R − , th e sum of the eleme nts in X ( α ) is equal to k α µ . By constraint (1 7), we h av e X α ∈ R − µ · τ ( α ) i,j, k σ j, k ≤ µ for all i ∈ I j . W e n ow apply the result o f Propo sition 3.1 with Γ = I j , M = µ and with the sets X ( α ) defined as above (here 3 N = d j ). Set the variables { w a : a ∈ C ( k ) Γ } ac cording to Proposition 3.1. Next, for k ∈ T j , we show h ow to define the variables { w ′ b : b ∈ C j , κ j ( b ) = k } . Initially , we set w ′ b = 0 for all b ∈ C j , κ j ( b ) = k . Ob serve that the values µ · z ( β ) i,j, k /σ j, k are non-negative integers for every i ∈ I , j ∈ J , k ∈ T j , β ∈ R − . For ev ery a ∈ C ( k ) Γ , we define w a words b (1) , b (1) , · · · , b ( w a ) ∈ C j . Assume some ordering on the elements β ∈ R − satisfying β H j,i = a i , namely β 1 , β 2 , · · · , β ℓ 0 for some positi ve integer ℓ 0 . For i ∈ I j , b ( ℓ ) i ( ℓ = 1 , 2 , · · · , w a ) is defined a s follows: b ( ℓ ) i is eq ual to β 1 for the first µ · z ( β 1 ) i,j, k /σ j, k words b (1) , b (2) , · · · , b ( w a ) ; b ( ℓ ) i is eq ual to β 2 for the ne xt µ · z ( β 2 ) i,j, k /σ j, k words, and so on . For every b ∈ C j we define w ′ b =    n i ∈ { 1 , 2 , · · · , w a } : b ( i ) = b o    . Finally , for e very b ∈ C j , κ j ( b ) = k , we d efine w j, b = σ j, k µ · w ′ b . Using Pr oposition 3.1, X a ∈C ( k ) Γ , a i = α w a = µ · τ ( α ) i,j, k σ j, k = X β : β H j,i = α µ · z ( β ) i,j, k σ j, k , and so all b (1) , b (2) , · · · , b ( w a ) (for all a ∈ C ( k ) Γ ) are well-defined. It is also straightforward to see that b ( ℓ ) ∈ C j for ℓ = 1 , 2 , · · · , w a . Next, we check th at th e newly- defined w j, b satisfy (3)-(5) f or e very j ∈ J , b ∈ C j . It is easy to see th at w j, b ≥ 0 ; th erefore (3) holds. By Proposition 3.1 we obtain σ j, k = X b ∈C j , κ j ( b )= k w j, b , for all j ∈ J , k ∈ T j , and τ ( α ) i,j, k = X b ∈C j , κ j ( b )= k , b i H j,i = α w j, b , for all j ∈ J , i ∈ I j , k ∈ T j , α ∈ R − . Let β H j,i = α . By the d efinition of w j, b it follows that X b ∈C j , κ ( b )= k , b i = β w j, b = z ( β ) i,j, k τ ( α ) i,j, k · X b ∈C j , κ ( b )= k , b i H j,i = α w j, b = z ( β ) i,j, k , where the first equality is due to th e definition o f the words b ( ℓ ) , ℓ = 1 , 2 , · · · , w a . By constraint ( 8) w e hav e, f or all j ∈ J , 1 = X k ∈T j σ j, k = X k ∈T j X b ∈C j , κ j ( b )= k w j, b = X b ∈C j w j, b , thus satisfy ing (4). Finally , by co nstraint (7) we obtain , for all j ∈ J , i ∈ I j , β ∈ R − , f ( β ) i = X k ∈T j z ( β ) i,j, k = X k ∈T j X b ∈C j , κ j ( b )= k , b i = β w j, b = X b ∈C j , b i = β w j, b , thus satisfy ing (5). Sketch of the Pr oof of Pr op osition 3. 1 In this proo f, we use a network flow app roach (see [6] for backgr ound material). The pro of will be by indu ction on M . W e set w a = 0 for all a ∈ C ( k ) Γ . W e show that there exists a vector a = ( a i ) i ∈ Γ ∈ C ( k ) Γ such that (i) For e very i ∈ Γ and α ∈ R − , a i = α = ⇒ x ( α ) i > 0 . (ii) If fo r some i ∈ Γ , P α ∈ R − x ( α ) i = M , then a i = α for some α ∈ R − . Then, we ‘ update’ th e values of x ( α ) i ’ s and M as follows. For every i ∈ Γ and α ∈ R − with a i = α we set x ( α ) i ← x ( α ) i − 1 . In add ition, we set M ← M − 1 . W e also set w a ← w a + 1 . It is e asy to see that the ‘u pdated’ values of x ( α ) i ’ s and M satisfy X i ∈ Γ x ( α ) i = k α M for all α ∈ R − , and P α ∈ R − x ( α ) i ≤ M for all i ∈ Γ . Therefo re, the ind uctive step can be applied with respect to these new values. Th e indu ction e nds when the value of M is equal to zer o. It is straightforward to see that when the ind uction termi- nates, ( 20) an d (21) hold with respect to the original values of the x ( α ) i and M . Existence of a th at satisfies (i): W e con struct a flow n etwork G = ( V , E ) as fo llows: V = { s, t } ∪ U 1 ∪ U 2 , where U 1 = R − and U 2 = Γ . Also set E = { ( s, α ) } α ∈ R − ∪ { ( i, t ) } i ∈ Γ ∪ { ( α, i ) } x ( α ) i > 0 . W e d efine a n integral capacity fun ction c : E − → N ∪ { + ∞} as follows: c ( e ) =    k α if e = ( s, α ) , α ∈ R − 1 if e = ( i, t ) , i ∈ Γ + ∞ if e = ( α, i ) , α ∈ R − , i ∈ Γ . (22) Next, apply the Ford-Fulkerson algorithm on the network ( G ( E , V ) , c ) to produc e a maximal flow f max . Since all the values o f c ( e ) a re integral for all e ∈ E , s o the values of f max ( e ) must all be integral for e very e ∈ E (see [6]). It can be shown that the minimum cu t in this graph has capacity c min = P α ∈ R − k α . 4 The flow f max in G has a value of P α ∈ R − k α . Observe tha t f max (( α, i )) ∈ { 0 , 1 } for all α ∈ R − and i ∈ Γ . Then, for all i ∈ Γ , we d efine a i =  α if f max (( α, i )) = 1 for some α ∈ U 1 0 other wise . For this selection of a = ( a 1 , a 2 , · · · , a N ) , we h av e a ∈ C ( k ) Γ and a i = α only if x ( α ) i > 0 . Existence o f a th at satisfies (i) an d (ii) simultaneo usly: W e start with the fo llowing definition. Definition 3.1: The v ertex i ∈ U 2 is called a c ritical vertex, if P α ∈ R − x ( α ) i = M . In o rder to have (19) satisfied after the next in ductive step, we have to d ecrease the value of P α ∈ R − x ( α ) i by (exactly) 1 for ev ery critical vertex. This is equi valent to having f max (( i, t )) = 1 . W e aim to show that there e xists a flo w f ∗ of the same value, which has f ∗ (( i, t )) = 1 for e very c ritical vertex i . Suppose that there is no such flow . Then, consider the maximu m flow f ′ , which h as f ′ (( i, t )) = 1 for the ma ximal possible number of the critical vertice s i ∈ U 2 . W e assum e that the re is a critical vertex i 0 ∈ U 2 , which ha s f ′ (( i 0 , t )) = 0 . It is possible to show that the flow f ′ can be mod ified towards the flow f ′′ of the same value, such that for f ′′ the nu mber of critical vertices i ∈ U 2 having f ′′ (( i, t )) = 1 is strictly larger than f or f ′ . It fo llows that there exists an integral flow f ∗ in ( G ( V , E ) , c ) of value P α ∈ R − k α , such that for every critical vertex i ∈ U 2 , f ∗ (( i, t )) = 1 . W e define a i =  α if f ∗ (( α, i )) = 1 f or some α ∈ U 1 0 otherwise . and a = ( a i ) i ∈ Γ . For this selection of a , we h av e a ∈ C ( k ) Γ and the p roperties (i) and ( ii) ar e satisfied. I V . C A S C A D E D P O LY T O P E R E P R E S E N TA T I O N In this section we show that the “cascaded p olytope ” representatio n described in [4] an d [5] can be exten ded to non- binary codes in a straightf orward manner . Below , we elaborate on the details. For j ∈ J , consid er th e j -th row H j of the p arity-check matrix H over R , and r ecall that C j = n ( b i ) i ∈I j : X i ∈I j b i · H j,i = 0 o . Assume that I j = { i 1 , i 2 , · · · , i d j } and den ote L j = { 1 , 2 , · · · , d j − 3 } . W e introd uce ne w variables χ j = ( χ j i ) i ∈L j and denote χ = ( χ j ) j ∈J . W e define a new linear code C ( χ ) j of length 2 d j − 3 by ( d j − 2) × (2 d j − 3) parity-ch eck matrix associated with the following set of parity-check eq uations over R : 1) b i 1 H j,i 1 + b i 2 H j,i 2 + χ j 1 = 0 . (23) 2) For ev ery ℓ = 1 , 2 , · · · , d j − 4 , − χ j ℓ + b i ℓ +2 H j,i ℓ +2 + χ j ℓ +1 = 0 . (24) 3) − χ j d j − 3 + b i d j − 1 H j,i d j − 1 + b i d j H j,i d j = 0 . (25) W e a lso define a linear cod e C ( χ ) of len gth n + P j ∈J ( d j − 3) defined by ( P j ∈J ( d j − 2)) × ( n + P j ∈J ( d j − 3)) par ity- check matr ix F associated with all the sets of parity -check equations (23)-(25) (f or all j ∈ J ). Theor em 4. 1: T he vector ( b i ) i ∈I j ∈ R d j is a codeword of C j if and only if ther e exists some vector χ j ∈ R d j − 3 such that (( b i ) i ∈I j | χ j ) ∈ C ( χ ) j . W e d enote by S the polyto pe corre sponding to th e LP relaxation prob lem (3)-(5) for the code C ( χ ) with the p arity- check matrix F . Let ( b , χ ) be a word in C ( χ ) , wh ere b ∈ C . It is natural to represent points in S as (( f , h ) , z ) , where f = ( f ( α ) i ) i ∈I , α ∈ R − and h = ( h ( α ) j,i ) j ∈J , i ∈L j , α ∈ R − are vectors of indicators cor respond ing to the entries b i ( i ∈ I ) in b and χ j i ( j ∈ J , i ∈ L j ) in χ , respectively . Theor em 4. 2: T he set ¯ S = { f : ∃ h , z s.t. (( f , h ) , z ) ∈ S } is equa l to the set ¯ Q = { f : ∃ w s.t. ( f , w ) ∈ Q} , an d therefor e, optimizing the linear cost function (2) over S is equiv alent to optimizing it over Q . It fo llows from Theore m 4.2 that the polyto pe S equ iv- alently describes the code C . This description h as at most n + m · ( d − 3) variables and m · ( d − 2) parity -check equ ations. Howe ver, the nu mber of variables participating in every parity- check equation is at most 3 . T herefor e, the total nu mber of variables and of equatio ns in the respective LP pr oblem will be b ound ed from above b y ( n + m ( d − 3))( q − 1) + m ( d − 2) · q 2 and m ( d − 2)( q 2 + 3 q − 2) . The polyto pe repre sentation in this section, whe n used with the LP pro blem in [3], lead s to a po lynomial- time decod er fo r a wide variety o f classical non -binary cod es. Its perform ance under LP decoding is y et to be studied. A C K N O W L E D G E M E N T S The auth ors wish to thank O. M ilenkovic f or interesting discussions. Th is work was supp orted by the Claude Shannon Institute for Discrete Mathematics, Coding and Crypto graphy (Science Foundation Irelan d Grant 06/MI/0 06). R E F E R E N C E S [1] J . F E L D M A N , Decoding Error- Corr ecting Codes via Linear Pr ogr am- ming, Ph.D. T hesis, Massachusett s Institute of T echnology , Sep. 2003. [2] J . F E L D M A N , M . J . W A I N W R I G H T , D . R . K A R G E R , Using linear pro- gramming to decode binary linear codes, IEEE T rans. Inform. Theory , vol. 51, no. 3, pp. 954–972, Mar . 2005. [3] M . F. F L A N A G A N , V . S K A C H E K , E . B Y R N E , M . G R E F E R AT H , Linear - programming Dec oding of Non-binary Linear Codes , Proc. 7th Int. ITG Confer . on Sourc e and Channel Coding (SCC) , Jan. 2008, Ulm, Germany . A vaila ble also at http://arxiv. org/abs/0707.43 60 . [4] M . C H E RT K OV , M . S T E PA N OV , Pseudo-code word Landscape, Pr oc. IEEE Int. Symp. on Inform. 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