Extended MinSum Algorithm for Decoding LDPC Codes over GF ( q )

Low density parity check (LDPC) codes designed over GF(q) (also referred to as GF(q)-LDPC codes) have been shown to approach Shannon limit performance for q = 2 and very long code lengths. On the other hand, for moderate code lengths, the error performance can be improved by increasing q [1]. However this improvement is achieved at the expense of increased decoding complexity. Simplified iterative decodings of GF(q)-LDPC codes have been investigated. For q = 2, the min-sum (MS) algorithm with proper modification has been shown to result in negligible performance degradation (less than 0.1 dB for regular LDPC codes) while performing additions only, and becoming independent of the channel conditions. Extension of this approach to any value q seems highly attractive. In [5], the MS algorithm is extended to any finite field of order q. Although only additions are performed, its complexity remains O(q). As a result, only small values of q can be considered by this algorithm and for q = 8, a degradation of 0.5 dB over BP decoding is reported. This performance gap increases with q. In this paper, we develop a generalization of the MS algorithm which not only performs additions without the need of channel estimation, but also with the two following objectives: (i) a complexity much lower than O(q) so that finite fields of large order can be considered; and (ii) a small performance degradation compared with BP decoding. The first objective is achieved by introducing configuration sets, which allow to keep only a small number of meaningful values at the check node processing. The second objective is achieved by applying at the variable node processing the correction techniques of [4] to the proposed algorithm.