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Designs Codes and Cryptography

Decoding error probability of random parity-check matrix ensemble over the erasure channel

Chin Hei Chan, Fang‐Wei Fu, Maosheng Xiong

Published Oct 16, 2024• 0 citations
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Abstract

Abstract In this paper we carry out an in-depth study on the average decoding error probability of the random parity-check matrix ensemble over the erasure channel under three decoding principles, namely unambiguous decoding, maximum likelihood decoding and list decoding. We obtain explicit formulas for the average decoding error probabilities of the random parity-check matrix ensemble under these three decoding principles and compute the error exponents. Moreover, for unambiguous decoding, we compute the variance of the decoding error probability of the random parity-check matrix ensemble and the error exponent of the variance, which implies a strong concentration result, that is, roughly speaking, the ratio of the decoding error probability of a random linear code in the ensemble and the average decoding error probability of the ensemble converges to 1 with high probability when the code length goes to infinity.

DOI
10.1007/s10623-024-01516-5

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Abstract In this paper we carry out an in-depth study on the average decoding error probability of the random parity-check matrix ensemble over the erasure channel under three decoding principles, namely unambiguous decoding, maximum likelihood decoding and list decoding. We obtain explicit formulas for the average decoding error probabilities of the random parity-check matrix ensemble under these three decoding principles and compute the error exponents. Moreover, for unambiguous decoding, we compute the variance of the decoding error probability of the random parity-check matrix ensemble and the error exponent of the variance, which implies a strong concentration result, that is, roughly speaking, the ratio of the decoding error probability of a random linear code in the ensemble and the average decoding error probability of the ensemble converges to 1 with high probability when the code length goes to infinity.
In this paper, we provide explicit formulas for the average decoding error probabilities of the ensemble of all (n,M)q(n,M)_q(n,M)q​ codes over the erasure channel under unambiguous, list and maximum-likelihood decoding principles. Moreover, we derive the asymptotic error exponents for the ensemble of all (n,qRn)q(n,q^{Rn})_q(n,qRn)q​ codes.
In this paper, we study the decoding error probability of linear codes over the erasure channel under the list decoding. The notion of the qℓ-incorrigible sets of linear codes is introduced to characterize its decoding error probability under the list decoding or the maximum likelihood decoding. By calculating the qℓ-incorrigible set distributions, the decoding error probability of a linear code over the erasure channel under the list decoding or the maximum likelihood decoding is expressed by its support weight distributions. For the ensemble of all [n,k] linear codes, the average decoding error probability under the maximum likelihood decoding and the average unsuccessful decoding probability under unambiguous decoding are determined. Furthermore, the error exponent of the average unsuccessful decoding probability under the unambiguous decoding is determined for the ensemble of all [n,nR] linear codes.
An efficient decoding algorithm for low-density parity-check (LDPC) codes on erasure channels with sporadic errors (i.e., binary error-and-erasure channels with error probability much smaller than the erasure probability) is proposed and its performance analyzed. A general single-error multiple-erasure (SEME) decoding algorithm is first described, which may be in principle used with any binary linear block code. The algorithm is optimum whenever the non-erased part of the received word is affected by at most one error, and is capable of performing error detection of multiple errors. An upper bound on the average block error probability under SEME decoding is derived for the linear random code ensemble. The bound is tight and easy to implement. The algorithm is then adapted to LDPC codes, resulting in a simple modification to a previously proposed efficient maximum likelihood LDPC erasure decoder which exploits the parity-check matrix sparseness. Numerical results reveal that LDPC codes under efficient SEME decoding can closely approach the average performance of random codes.
In this paper, we study the list decoding error probability of a linear code over the erasure channel. The notion of L-incorrigible sets of a linear code is introduced to characterize its performance under list decoding. The L-incorrigible set distribution of a linear code can also be used to completely determine its decoding error probability under maximum likelihood decoding over the erasure channel. Furthermore, we show that the L-incorrigible set distribution of a linear code can be determined by its support weight distribution. Finally, the error exponent of the unsuccessful decoding probability under optimal decoding for the ensemble of all [n, nR] linear codes is determined.
In this paper, we are concerned with the finite-length analysis of low-density parity-check (LDPC) codes when used over the binary erasure channel (BEC). The main result is an expression for the exact average bit and block erasure probability for a given regular ensemble of LDPC codes when decoded iteratively. We also give expressions for upper bounds on the average bit and block erasure probability for regular LDPC ensembles and the standard random ensemble under maximum-likelihood (ML) decoding. Finally, we present what we consider to be the most important open problems in this area.
We derive both upper and lower bounds on the decoding error probability of maximum-likelihood (ML) decoded low-density parity-check (LDPC) codes. The results hold for any binary-input symmetric-output channel. Our results indicate that for various appropriately chosen ensembles of LDPC codes, reliable communication is possible up to channel capacity. However, the ensemble averaged decoding error probability decreases polynomially, and not exponentially. The lower and upper bounds coincide asymptotically, thus showing the tightness of the bounds. However, for ensembles with suitably chosen parameters, the error probability of almost all codes is exponentially decreasing, with an error exponent that can be set arbitrarily close to the standard random coding exponent.