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LDPC codes which can correct three errors under iterative decoding

Anantha Raman Krishnan, Bane V. Vasic

Published Jan 1, 2008• 19 citations
OpenAlex

Abstract

Abstract — In this paper, we provide necessary and sufficient conditions for a column-weight-three LDPC code to correct three errors when decoded using Gallager A algorithm. We then provide a construction technique which results in a code satisfying the above conditions. We also provide numerical assessment of code performance via simulation results. I.

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LDPC codes which can correct three errors under iterative decoding

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Bane Vasic, et al.

In this paper, we investigate the error correction capability of column-weight-three LDPC codes when decoded using the Gallager A algorithm. We prove that the necessary condition for a code to correct k ≥ 5 errors is to avoid cycles of length up to 2k in its Tanner graph. As a consequence of this result, we show that given any α> 0, ∃N such that ∀n> N, no code in the ensemble of column-weight-three codes can correct all αn or fewer errors. We extend these results to the bit flipping algorithm.

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Error Correction Capability of Column-Weight-Three LDPC Codes Under the Gallager A Algorithm—Part II

Shashi Kiran Chilappagari, Dung Viet Nguyen, Bane V. Vasic +1 more

The relation between the girth and the error correction capability of column-weight-three LDPC codes under the Gallager A algorithm is investigated. It is shown that a column-weight-three LDPC code with Tanner graph of girthg¿ 10 can correct all error patterns with up to(g/2-1) errors in at mostg/2 iterations of the Gallager A algorithm. For codes with Tanner graphs of girthg¿ 8, it is shown that girth alone cannot guarantee correction of all error patterns with up to(g/2-1) errors under the Gallager A algorithm. Sufficient conditions to correct(g/2-1) errors are then established by studying trapping sets.

IEEE Transactions on Information Theory•May 27, 2010•20 citations
Abstract — In this paper, we provide necessary and sufficient conditions for a column-weight-three LDPC code to correct three errors when decoded using Gallager A algorithm. We then provide a construction technique which results in a code satisfying the above conditions. We also provide numerical assessment of code performance via simulation results. I.
In this paper, we give necessary and sufficient conditions for low-density parity-check (LDPC) codes with column-weight three to correct three errors when decoded using hard-decision message-passing decoding. Additionally, we give necessary and sufficient conditions for column-weight-four codes to correct three errors in four iterations of hard-decision message-passing decoding. We then give a construction technique which results in codes satisfying these conditions. We also provide numerical assessment of code performance via simulation results.
In this paper, the error-correction capability of column-weight-three low-density parity-check (LDPC) codes when decoded using the Gallager A algorithm is investigated. It is proved that a necessary condition for a code to correct all error patterns with up to k ges 5 errors is to avoid cycles of length up to 2k in its Tanner graph. As a consequence of this result, it is shown that given any alpha > 0, exist N such that forall n > N, no code in the ensemble of column-weight-three codes can correct all alphan or fewer errors. The results are extended to the bit flipping algorithms.
In this paper, we investigate the error correction capability of column-weight-three LDPC codes when decoded using the Gallager A algorithm. We prove that the necessary condition for a code to correct k≥5k \geq 5k≥5 errors is to avoid cycles of length up to 2k2k2k in its Tanner graph. As a consequence of this result, we show that given any $α>0, \exists N $ such that ∀n>N\forall n>N∀n>N, no code in the ensemble of column-weight-three codes can correct all αnαnαn or fewer errors. We extend these results to the bit flipping algorithm.
In this paper, we investigate the error correction capability of column-weight-three LDPC codes when decoded using the Gallager A algorithm. We prove that the necessary condition for a code to correct k ≥ 5 errors is to avoid cycles of length up to 2k in its Tanner graph. As a consequence of this result, we show that given any α> 0, ∃N such that ∀n> N, no code in the ensemble of column-weight-three codes can correct all αn or fewer errors. We extend these results to the bit flipping algorithm.
The relation between the girth and the error correction capability of column-weight-three LDPC codes under the Gallager A algorithm is investigated. It is shown that a column-weight-three LDPC code with Tanner graph of girthg¿ 10 can correct all error patterns with up to(g/2-1) errors in at mostg/2 iterations of the Gallager A algorithm. For codes with Tanner graphs of girthg¿ 8, it is shown that girth alone cannot guarantee correction of all error patterns with up to(g/2-1) errors under the Gallager A algorithm. Sufficient conditions to correct(g/2-1) errors are then established by studying trapping sets.