Binary representation of cycle Tanner-graph GF(2b) codes

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^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^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^b) with a block-error rate <10^-4 at E_b/N₀=1.89 dB, which appears to exhibit the best iterative-decoding performance at this short block length known to date.