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

Yoshiyuki Kabashima, Naoya Sazuka A, K. Nakamura +1 more

The replica method, developed in statistical physics, is employed in conjunction with Gallager's methodology to accurately evaluate zero error noise thresholds for Gallager code ensembles. Our approach generally provides more optimistic evaluations than those reported in the information theory literature for sparse matrices; the difference vanishes as the parity check matrix becomes dense.

Tadashi Wadayama

Yoshiyuki Kabashima, Naoya Sazuka A, Kazutaka Nakamura +1 more

Statistical physics is employed to evaluate the performance of error-correcting codes in the case of finite message length for an ensemble of Gallager's error correcting codes. We follow Gallager's approach of upper bounding the average decoding error rate, but invoke the replica method to reproduce the tightest general bound to date, and to improve on the most accurate zero-error noise level threshold reported in the literature. The relation between the methods used and those presented in the information theory literature are explored.

Nicolas Sourlas

祥介 樺島, Yoshiyuki Kabashima

Sheng Tong

An efficient numerical approach to the exact ensemble distance spectrum of Gallager codes has been developed by evaluating powers of polynomials. With the exact ensemble distance spectrum of Gallager codes, tangential sphere upper bounds on their maximum likelihood (ML) decoding performance over binary input AWGN channels are investigated. Numerical results indicate improved bounds have been obtained, better than Sason and Shamai's results (which are based on Gallager's upper bound on the ensemble distance spectrum), especially in the error floor region. Furthermore, some critical properties of Gallager codes, including typical minimum distance and the performance tradeoff in the waterfall and error floor regions, have been considered.