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

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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

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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

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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

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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

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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

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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

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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.

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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

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