Statistical mechanics of error exponents for error-correcting codes

Details Statistical mechanics of error exponents for error-correcting codes Statistical mechanics of error exponents for error-correcting codes

2006-06-27

arxiv.org/abs/cond-mat/0606696v1

Phys. Rev. E 74, 056110 (2006)

Physics

Condensed Matter

Disordered Systems and Neural Networks

32 pages, 13 figures

Scientific paper

10.1103/PhysRevE.74.056110

Error exponents characterize the exponential decay, when increasing message length, of the probability of error of many error-correcting codes. To tackle the long standing problem of computing them exactly, we introduce a general, thermodynamic, formalism that we illustrate with maximum-likelihood decoding of low-density parity-check (LDPC) codes on the binary erasure channel (BEC) and the binary symmetric channel (BSC). In this formalism, we apply the cavity method for large deviations to derive expressions for both the average and typical error exponents, which differ by the procedure used to select the codes from specified ensembles. When decreasing the noise intensity, we find that two phase transitions take place, at two different levels: a glass to ferromagnetic transition in the space of codewords, and a paramagnetic to glass transition in the space of codes.

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