Typical performance of Gallager-type error-correcting codes

祥介 樺島, Yoshiyuki Kabashima

祥介 樺島, YOSHIYUKI KABASHIMA

Yoshiyuki Kabashima, Tatsuto Murayama, David Saad

The performance of Gallager's error-correcting code is investigated via methods of statistical physics. In this approach, the transmitted codeword comprises products of the original message bits selected by two randomly constructed sparse matrices; the number of nonzero row/column elements in these matrices constitutes a family of codes. We show that Shannon's channel capacity is saturated for many of the codes while slightly lower performance is obtained for others which may be of higher practical relevance. Decoding aspects are considered by employing the Thouless-Anderson-Palmer approach which is identical to the commonly used belief-propagation-based decoding.

Tatsuto Murayama, Yoshiyuki Kabashima, David Saad +1 more

A variation of Gallager error-correcting codes is investigated using statistical mechanics. In codes of this type, a given message is encoded into a codeword that comprises Boolean sums of message bits selected by two randomly constructed sparse matrices. The similarity of these codes to Ising spin systems with random interaction makes it possible to assess their typical performance by analytical methods developed in the study of disordered systems. The typical case solutions obtained via the replica method are consistent with those obtained in simulations using belief propagation decoding. We discuss the practical implications of the results obtained and suggest a computationally efficient construction for one of the more practical configurations.

Richard W. Hamming

Ido Kanter, David Saad

Gallager-type error-correcting codes that nearly saturate Shannon's bound are constructed using insight gained from mapping the problem onto that of an Ising spin system. The performance of the suggested codes is evaluated for different code rates for both finite and infinite message lengths.