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List decoding of convolutional codes over integer residue rings


Lieb, Julia; Napp, Diego; Pinto, Raquel (2021). List decoding of convolutional codes over integer residue rings. Finite Fields and Their Applications, 72:101815.

Abstract

A convolutional code Cover Zpr[D]is a Zpr[D]-submodule of Znpr[D]where Zpr[D] stands for the ring of polynomials with coefficients in Zpr. In this paper, we study the list decoding problem of these codes when the transmission is performed over an erasure channel, that is, we study how much information one can recover from a codeword w∈Cwhen some of its coefficients have been erased. We do that using the p-adic expansion of wand particular representations of the parity-check polynomial matrix of the code. Fr o m these matrix polynomial representations we recursively select certain equations that wmust satisfy and have only coefficients in the field pr−1Zpr. We exploit the natural block Toeplitz structure of the sliding parity-check matrix to derive a step by step methodology to obtain a list of possible codewords for a given corrupted codeword w, that is, a list with the closest codewords to w.

Abstract

A convolutional code Cover Zpr[D]is a Zpr[D]-submodule of Znpr[D]where Zpr[D] stands for the ring of polynomials with coefficients in Zpr. In this paper, we study the list decoding problem of these codes when the transmission is performed over an erasure channel, that is, we study how much information one can recover from a codeword w∈Cwhen some of its coefficients have been erased. We do that using the p-adic expansion of wand particular representations of the parity-check polynomial matrix of the code. Fr o m these matrix polynomial representations we recursively select certain equations that wmust satisfy and have only coefficients in the field pr−1Zpr. We exploit the natural block Toeplitz structure of the sliding parity-check matrix to derive a step by step methodology to obtain a list of possible codewords for a given corrupted codeword w, that is, a list with the closest codewords to w.

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Additional indexing

Item Type:Journal Article, not_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 > Algebra and Number Theory
Physical Sciences > General Engineering
Physical Sciences > Applied Mathematics
Uncontrolled Keywords:Applied Mathematics, General Engineering, Algebra and Number Theory, Theoretical Computer Science
Language:English
Date:1 June 2021
Deposited On:10 Nov 2021 14:00
Last Modified:26 Apr 2024 01:36
Publisher:Elsevier
ISSN:1071-5797
OA Status:Hybrid
Free access at:Publisher DOI. An embargo period may apply.
Publisher DOI:https://doi.org/10.1016/j.ffa.2021.101815
Project Information:
  • : FunderDeutsche Forschungsgemeinschaft
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  • : FunderMinisterio de Ciencia e Innovación
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  • : FunderFundação para a Ciência e a Tecnologia
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  • : FunderSchweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung
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  • Content: Published Version
  • Language: English
  • Licence: Creative Commons: Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0)