Symbol Error Rate of Binary Block Codes

Gérard Cohen, Philippe Godlewski

Michele Elia

X.-Yu. Hu, Evangelos S. Eleftheriou

We derive the average symbol and Hamming weight spectrum functions of the random ensemble of regular low-density parity-check (LDPC) codes over GF(2/sup b/) when used with the binary-input noisy channel. This work confirms theoretically that the near-Shannon-limit performance of Gallager's binary LDPC codes can be significantly enhanced by moving to fields of higher order. We construct a family of error-correcting codes based on the binary representation of GF(2/sup b/) codes defined on a cycle Tanner graph that appears to be "good" for both optimum and iterative decoding over the binary-input noisy channel. In particular, we report a short-block-length (1008 bits), rate-1/2 progressive-edge-growth-based cycle Tanner-graph code over GF(2/sup b/) with a block-error rate <10/sup -4/ at E/sub b//N/sub 0/=1.89 dB, which appears to exhibit the best iterative-decoding performance at this short block length known to date.

Don J. Torrieri

There are two types of bounded-distance decoders for linear block codes: erasing decoders that discard uncorrectable received words, and reproducing decoders that reproduce uncorrectable received words. Exact expressions for the information-symbol and decoded-symbol error rates are derived for both types. Necessary and sufficient conditions are derived for the quality of the information-symbol and decoded symbol error rates, It is formally proved that these two error rates are equal for cyclic codes with either erasing or reproducing decoders. For reproducing decoders, two approximations to the information-bit error rate and their applicability are examined.>

Werner Henkel

J.T. Coffey

Decoding rules for linear block codes used over binary symmetric channels with crossover probability p when the goal is to minimize the probability that sub-blocks of information bits are in error are considered, with application to the (24,12,8) Golay code. (The optimal rules to minimize the probability of individual bits being in error are known for all p.) The results require detailed knowledge of the weight enumerators of given sub codes, which can be calculated from knowledge of the orbit and coset structure of the code. It is interesting that in all cases examined, the rule in decoding a block given the coset received is to select the symbol that is optimal as p → 0 or the symbol that is optimal as p → 1/2, depending on p.