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Construction of LDPC convolutional codes via difference triangle sets


Alfarano, Gianira Nicoletta; Lieb, Julia; Rosenthal, Joachim (2021). Construction of LDPC convolutional codes via difference triangle sets. Designs, Codes and Cryptography, 89:2235-2254.

Abstract

In this paper, a construction of $(n,k,\delta )$ LDPC convolutional codes over arbitrary finite fields, which generalizes the work of Robinson and Bernstein and the later work of Tong is provided. The sets of integers forming a (k, w)-(weak) difference triangle set are used as supports of some columns of the sliding parity-check matrix of an $(n,k,\delta )$ convolutional code, where $n\in {\mathbb {N}}$, $n>k$. The parameters of the convolutional code are related to the parameters of the underlying difference triangle set. In particular, a relation between the free distance of the code and w is established as well as a relation between the degree of the code and the scope of the difference triangle set. Moreover, we show that some conditions on the weak difference triangle set ensure that the Tanner graph associated to the sliding parity-check matrix of the convolutional code is free from $2\ell $-cycles not satisfying the full rank condition over any finite field. Finally, we relax these conditions and provide a lower bound on the field size, depending on the parity of $\ell $, that is sufficient to still avoid $2\ell $-cycles. This is important for improving the performance of a code and avoiding the presence of low-weight codewords and absorbing sets.

Abstract

In this paper, a construction of $(n,k,\delta )$ LDPC convolutional codes over arbitrary finite fields, which generalizes the work of Robinson and Bernstein and the later work of Tong is provided. The sets of integers forming a (k, w)-(weak) difference triangle set are used as supports of some columns of the sliding parity-check matrix of an $(n,k,\delta )$ convolutional code, where $n\in {\mathbb {N}}$, $n>k$. The parameters of the convolutional code are related to the parameters of the underlying difference triangle set. In particular, a relation between the free distance of the code and w is established as well as a relation between the degree of the code and the scope of the difference triangle set. Moreover, we show that some conditions on the weak difference triangle set ensure that the Tanner graph associated to the sliding parity-check matrix of the convolutional code is free from $2\ell $-cycles not satisfying the full rank condition over any finite field. Finally, we relax these conditions and provide a lower bound on the field size, depending on the parity of $\ell $, that is sufficient to still avoid $2\ell $-cycles. This is important for improving the performance of a code and avoiding the presence of low-weight codewords and absorbing sets.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:340 Law
610 Medicine & health
510 Mathematics
Scopus Subject Areas:Physical Sciences > Theoretical Computer Science
Physical Sciences > Computer Science Applications
Physical Sciences > Discrete Mathematics and Combinatorics
Physical Sciences > Applied Mathematics
Uncontrolled Keywords:Applied Mathematics, Computer Science Applications
Language:English
Date:22 July 2021
Deposited On:25 Aug 2021 12:59
Last Modified:25 Feb 2024 02:42
Publisher:Springer
ISSN:0925-1022
OA Status:Hybrid
Publisher DOI:https://doi.org/10.1007/s10623-021-00912-5
Project Information:
  • : FunderSNSF
  • : Grant ID200021_188430
  • : Project TitleNew Constructions of Convolutional Codes
  • Content: Published Version
  • Language: English
  • Licence: Creative Commons: Attribution 4.0 International (CC BY 4.0)