Statistical Mechanics of Low-Density Parity Check Error-Correcting Codes over Galois Fields

A variation of low density parity check (LDPC) error correcting codes defined over Galois fields ($GF(q)$) is investigated using statistical physics. A code of this type is characterised by a sparse random parity check matrix composed of $C$ nonzero elements per column. We examine the dependence of the code performance on the value of $q$, for finite and infinite $C$ values, both in terms of the thermodynamical transition point and the practical decoding phase characterised by the existence of a unique (ferromagnetic) solution. We find different $q$-dependencies in the cases of C=2 and $C \ge 3$; the analytical solutions are in agreement with simulation results, providing a quantitative measure to the improvement in performance obtained using non-binary alphabets.Comment: 7 pages, 1 figur

Evaluating zero error noise thresholds by the replica method for Gallager code ensembles

The zero error noise threshold for Gallager code ensemblers were evaluated by using replica method (RM). The RM, which was invented in statistical physics, offered an option for calculating the bound. It was shown that the given approach provided more optimistic evaluations with respect to the evaluations provided by the information theory literature for sparse matrices