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IEEE Transactions on Information Theory

Errata for "good error-correcting codes based on very sparse matrices"

David Mackay

Published Jul 1, 2001• 13 citations
Publisher

Abstract

Prospective authors are requested to submit new, unpublished manuscripts for inclusion in the upcoming event described in this call for papers.
DOI
10.1109/tit.2001.930952

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We study two families of error-correcting codes defined in terms of very sparse matrices. "MN" (MacKay-Neal (1995)) codes are recently invented, and "Gallager codes" were first investigated in 1962, but appear to have been largely forgotten, in spite of their excellent properties. The decoding of both codes can be tackled with a practical sum-product algorithm. We prove that these codes are "very good", in that sequences of codes exist which, when optimally decoded, achieve information rates up to the Shannon limit. This result holds not only for the binary-symmetric channel but also for any channel with symmetric stationary ergodic noise. We give experimental results for binary-symmetric channels and Gaussian channels demonstrating that practical performance substantially better than that of standard convolutional and concatenated codes can be achieved; indeed, the performance of Gallager codes is almost as close to the Shannon limit as that of turbo codes.
This paper presents a new class of easily invertible circulant matrices, defined by exploiting the isomorphism from the ring Mnof n times n circulant matrices over GF(p) to the ring Rn= GF(p)[x]/(xn- 1) of the polynomials modulo (xn- 1). Such class contains matrices free of 4-length cycles that, if sparse, can be included in the parity check matrix of QC-LDPC codes. Bounds for the weight of their inverses are also determined, that are useful for designing sparse generator matrices for these error correcting codes.