On the effect of parity-check weights in iterative decoding

Santhi, Nandakishore; Vardy, Alexander

doi:10.1109/isit.2004.1365137

Cited by 9 publications

(14 citation statements)

References 4 publications

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“…The number of undecodable erasure patterns for a maximum-likelihood decoder and for the iterative decoders based on and is given in Table IV. (20) Proof: Assume to the contrary that there is a parity-check matrix for with and at most rows. As in Theorem 5, we say that a given set with is an -set, and that a row of covers an -set if the projection of on has weight one.…”

Section: Golay Codesmentioning

confidence: 99%

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On the stopping distance and the stopping redundancy of codes

Schwartz

Vardy

2005

Proceedings. International Symposium on Information Theory, 2005. ISIT 2005.

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173

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Abstract-It is now well known that the performance of a linear code under iterative decoding on a binary erasure channel (and other channels) is determined by the size of the smallest stopping set in the Tanner graph for . Several recent papers refer to this parameter as the stopping distance of . This is somewhat of a misnomer since the size of the smallest stopping set in the Tanner graph for depends on the corresponding choice of a parity-check matrix. It is easy to see that , where is the minimum Hamming distance of , and we show that it is always possible to choose a parity-check matrix for (with sufficiently many dependent rows) such that = . We thus introduce a new parameter, the stopping redundancy of , defined as the minimum number of rows in a parity-check matrix for such that the corresponding stopping distance ( ) attains its largest possible value, namely, ( ) = . We then derive general bounds on the stopping redundancy of linear codes. We also examine several simple ways of constructing codes from other codes, and study the effect of these constructions on the stopping redundancy. Specifically, for the family of binary Reed-Muller codes (of all orders), we prove that their stopping redundancy is at most a constant times their conventional redundancy. We show that the stopping redundancies of the binary and ternary extended Golay codes are at most 34 and 22, respectively. Finally, we provide upper and lower bounds on the stopping redundancy of MDS codes.

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Section: Golay Codesmentioning

confidence: 99%

“…In algebraic coding theory, a parity-check matrix for a linear code usually has linearly independent rows. However, in the context of iterative decoding, it has been already observed in [20], [24], and other papers that adding linearly dependent rows to can be advantageous. Certainly, this can increase the stopping distance .…”

Section: Introductionmentioning

confidence: 99%

On the stopping distance and the stopping redundancy of codes

Schwartz

Vardy

2005

Proceedings. International Symposium on Information Theory, 2005. ISIT 2005.

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173

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“…Let us consider the case ℓ − 1 = λ n, for 0 < λ < 1. In this case, the logarithm in the numerator grows at most linearly with n, since 22) and the second sum under the logarithm never exceeds the first. As a result, the asymptotic behavior of the given upper bound is dominated by the expression in the denominator, which, for large n, takes the form…”

Section: Appendix II Asymptotic Behavior Of the Bound In Theorem 38mentioning

confidence: 99%

“…Similarly as for the Golay code, we only consider minimum-weight codewords of the dual code for the purpose of generating redundant parity-check matrices in cyclic form [22]. In particular, we focus on four We now compare the stopping distance properties of parity-check matrices of cyclic form with that of a novel construction of redundant parity-check matrices generalizing the method in [10].…”

Section: ) the Binary Golaymentioning

confidence: 99%

Permutation Decoding and the Stopping Redundancy Hierarchy of Linear Block Codes

Hehn

Milenković

Laendner

et al. 2007

2007 IEEE International Symposium on Information Theory

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We introduce the notion of the stopping redundancy hierarchy of a linear block code as a measure of the trade-off between performance and complexity of iterative decoding for the binary erasure channel. We derive lower and upper bounds for the stopping redundancy hierarchy via Lovász's Local Lemma and Bonferroni-type inequalities, and specialize them for codes with cyclic parity-check matrices. Based on the observed properties of parity-check matrices with good stopping redundancy characteristics, we develop a novel decoding technique, termed automorphism group decoding, that combines iterative message passing and permutation decoding. We also present bounds on the smallest number of permutations of an automorphism group decoder needed to correct any set of erasures up to a prescribed size. Simulation results demonstrate that for a large number of algebraic codes, the performance of the new decoding method is close to that of maximum likelihood decoding.

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“…This is equivalent to adding redundant parity checks to the standard parity check matrix. The technique is also applied in soft decoding of some of the classical codes [26][27][28]. Fig.…”

Section: Graphical Models With Redundancymentioning

confidence: 99%

An Iterative Algorithm and Low Complexity Hardware Architecture for Fast Acquisition of Long PN Codes in UWB Systems

Yeung

Chugg

2006

J VLSI Sign Process Syst Sign Image Video Technol

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Abstract. Rapidly acquiring the code phase of the spreading sequence in an ultra-wideband system is a very difficult problem. In this paper, we present a new iterative algorithm and its hardware architecture in detail. Our algorithm is based on running iterative message passing algorithms on a standard graphical model augmented with multiple redundant models. Simulation results show that our new algorithm operates at lower signal to noise ratio than earlier works using iterative message passing algorithms. We also demonstrate an efficient hardware architecture for implementing the new algorithm. Specifically, the redundant models can be combined together so that substantial memory usage can be reduced. Our prototype achieves the cost-speed product unachievable by traditional approaches.

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On the effect of parity-check weights in iterative decoding

Cited by 9 publications

References 4 publications

On the stopping distance and the stopping redundancy of codes

On the stopping distance and the stopping redundancy of codes

Permutation Decoding and the Stopping Redundancy Hierarchy of Linear Block Codes

An Iterative Algorithm and Low Complexity Hardware Architecture for Fast Acquisition of Long PN Codes in UWB Systems

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