# (speaker) Fedor Groshev Vladimir Potapov Victor Zyablov IITP RAS, Moscow.

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(speaker) Fedor Groshev Vladimir Potapov Victor Zyablov IITP RAS, Moscow

 Reed-Solomon code-based LDPC (RS-LDPC) block codes are obtained by replacing single parity-check codes in Gallager’s LDPC codes with Reed-Solomon constituent codes.  This paper investigates asymptotic error correcting capabilities of ensembles of random RS-LDPC codes, used over the binary symmetric channel and decoded with a low- complexity hard-decision iterative decoding algorithm.  Estimation of the number decoding algorithm iterations.  It is shown that there exist RS-LDPC codes for which such iterative decoding corrects any error pattern with a number of errors that grows linearly with the code length.  The results are illustrated by numerical examples, for various choices of code parameters. 2

 Parity-check matrix of RS-LDPC codes H 0 :  (n 0, k 0, d 0 ) extended Reed-Solomon codes are constituent codes, d 0 = 3  l random column permutations of H b form layers of H  Code rate is 3

 Bipartite Tanner graph of RS- LDPC codes.  Constraint nodes have degree n 0 and represent constituent RS codes.  Constituent parity-check matrices H 0j,k are all equal up to column permutations.  Variable nodes have degree l and represent code symbols.  Each variable node is connected to exactly one constraint node in each layer. 4

Iterative hard-decision decoding algorithm, whose decoding iterations i = 1,2,…i max consist of the following two steps: 1) For the tentative sequence r (i), where r (1) is the received sequence r, decode independently all constituent RS codes ( lb), which can correct one error. If all the constituent codes have zero syndrome, then output v = r (i) and stop. Otherwise, proceed to step 2. 2) In the each sequence r (i), set to the new value all positions, which are corrected by constituent RS codes. This yields the next updated sequence r (i+1). If r (i+1) = r (i), declare the decoding failure and stop. Otherwise, return to step (1). 5

Lemma1: For decoding algorithm convergence required that, in each iteration the number of errors correctable by the constituent codes is larger than the number of insertion errors. Lemma2: If for any error pattern with  2/3+ , 0<  <1, then the number of correctable errors in any such error pattern is always larger than the number of uncorrectable errors. Lemma3: For any RS-LDPC code from the ensemble C(n 0, l, b), if in each iteration of the algorithm corrected fixed part of errors (  >2/3+  ), then algorithm yields a correct decision after O(log n) iterations, where n = bn 0 is the code length. 6

Theorem: In the ensemble C(n 0, l, b) of RS-LDPC codes, there exist codes (with probability p, where ), which can correct any error pattern of weight up to   n, with decoding complexity O(n log n). The value   is the largest root of the equation where and where  > 2/3 and the maximization is performed over all s such that 7 The proof is similar to proof in the work: V. V. Zyablov and M. S. Pinsker, “Estimation of the error-correction complexity for Gallager low-density codes”1975

Fig. 2. Values of   computed for  =0.67 according to Theorem for several code ensembles of different rates with the fixed constituent code length n 0 = 128. 8 Fig. 1. Values of   computed for  = 0.67 according to Theorem for several code ensembles of rates approximately R  1/2. The maximum is achieved with the constituent code length n 0 = 128.

 We have studied the performance of ensembles of Reed- Solomon code-based LDPC codes, with the distance of RS code d 0 =3, used over the BSC, when the code length n grows to infinity.  It was shown that these codes can be decoded with a simple iterative decoding algorithm whose complexity is O(n*log n).  Also was proved the existence of RS LDPC codes with the fixed constituent code distance (d 0 =3), Compare to the work of A. Barg and G. Zemor, “Error exponents of expander codes” 2002, where was shown estimation of constituent code distance  The maximum fraction of errors , correctable with the iterative decoder, was computed numerically for two types of code ensembles: codes of fixed rate R  1/2 and codes of variable rates with a fixed constituent Reed-Solomon code. 9