Structured LDPC codes over GF(2m) and companion matrix based decoding

It is well known that random-like low-density parity-check (LDPC) codes over the extension fields GF(2^m) of GF(2), for m>1, tend to outperform their binary counterparts of comparable length and rate. At the same time, structured LDPC codes offer the advantage of reduced implementation and storage complexity, so that it is of interest to investigate mathematical design methods for codes on graphs over fields of large order. We propose a new class of combinatorially developed codes obtained by properly combining Reed-Solomon (RS) type parity-check matrices and sparse parity-check matrices based on permutation matrices. The proposed codes have large girth and minimum distance. In order to further reduce the decoding complexity of the proposed scheme, we introduce a new decoding algorithm based on matrix representations of the underlying field, which trades performance for complexity. The particular field representation described in this abstract is based on a power basis generated by a companion matrix of a primitive polynomial of the field GF(2^m). It is observed that the choice of the primitive polynomial influences the cycle distribution of the code graph.