Approximate iterative LP decoding of LDPC codes over GF(q) in linear complexity

The problem of low-complexity approximated linear programming (LP) decoding of LDPC codes over GF(q) is considered. An iterative algorithm with linear complexity was proposed by Burshtein for the binary case. However, this algorithm cannot be trivially generalized for the non-binary case, since the derivation uses the even parity property of the check nodes, which does not hold for codes over GF(q). In this work the algorithm is generalized to the non-binary case. We show, that by applying this algorithm to a softened version of the non-binary LP problem proposed by Flanagan, Skachek, Byrne and Grefeath, we can obtain, with complexity linear in the block length, a feasible solution vector for the non-binary LP decoding problem, which is arbitrarily close to optimal in the following sense. The distance between the minimum value achieved by the exact LP decoder and the objective function value of the approximate solution, normalized by the block length of the code, can be made arbitrarily small. We present simulation results and comparisons with the belief propagation (BP) algorithm using-regular ternary LDPC codes. The proposed algorithm can be easily extended to generalized LDPC codes.