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

Kazutaka Nakamura·Yoshiyuki Kabashima·David Saad
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Abstract

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.

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Kazutaka Nakamura, Yoshiyuki Kabashima, David Saad. 2000-10-05. Statistical Mechanics of Low-Density Parity Check Error-Correcting Codes over Galois Fields. https://doi.org/10.1209/epl%2Fi2001-00564-y

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