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PreprintarXiv licencearXiv

Convolutional Codes from Cyclic Codes with Guaranteed Free and Local Minimum Distances

  • Khaled Abdel-Ghaffar
  • Daniel J. Costello Jr.
  • Juane Li
  • Shu Lin
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Abstract

This paper presents an algebraic method to construct convolutional codes with guaranteed \emph{free and local minimum distances} without limit based on cyclic codes of odd lengths. The constructions are simple but effective, and no computer search is needed. For any two positive integers r and t with 1 ≀ r < t, a rate-r/t convolutional code π’žconvol can be constructed by using a chain of r cyclic codes π’ž0, π’ž1, …, π’žrβˆ’1 of the same length n which satisfy the inclusion condition, π’ž0 βŠƒ π’ž1 βŠƒ … βŠƒ π’žrβˆ’1. Such a convolutional code π’žconvol is composed of a \emph{semi-infinite chain of identical local codes} confined in a diagonal band of width n. Each local code π’žlocal of π’žconvol is formed from the r cyclic codes in the code chain and is a specially localized subcode of the \emph{mother code} π’ž0 in the code chain. The minimum distance dlocal of each local code of π’žconvol is lower bounded by the minimum distance d0 of the mother code π’ž0 in the code chain. The local structure of π’žconvol allows it to be decoded based on a designed parity-check matrix of the mother code π’ž0 using a sliding window decoding scheme.

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Cite

BibTeXRIS
BibTeX
@misc{abdelghaffar2026convolutional,
  title = {{Convolutional Codes from Cyclic Codes with Guaranteed Free and Local Minimum Distances}},
  author = {Khaled Abdel-Ghaffar and Daniel J. Costello Jr. and Juane Li and Shu Lin},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.08296},
  eprint = {2609.08296},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.08296},
}
RIS
TY  - UNPB
TI  - Convolutional Codes from Cyclic Codes with Guaranteed Free and Local Minimum Distances
AU  - Khaled Abdel-Ghaffar
AU  - Daniel J. Costello Jr.
AU  - Juane Li
AU  - Shu Lin
PY  - 2026
DA  - 2026/09/08
PB  - arXiv
DO  - 10.48550/arxiv.2609.08296
UR  - https://arxiv.org/abs/2609.08296
AB  - This paper presents an algebraic method to construct convolutional codes with guaranteed \emph{free and local minimum distances} without limit based on cyclic codes of odd lengths. The constructions are simple but effective, and no computer search is needed. For any two positive integers r and t with 1 ≀ r < t, a rate-r/t convolutional code π’ž_convol can be constructed by using a chain of r cyclic codes π’ž_0, π’ž_1, …, π’ž_rβˆ’1 of the same length n which satisfy the inclusion condition, π’ž_0 βŠƒ π’ž_1 βŠƒ … βŠƒ π’ž_rβˆ’1. Such a convolutional code π’ž_convol is composed of a \emph{semi-infinite chain of identical local codes} confined in a diagonal band of width n. Each local code π’ž_local of π’ž_convol is formed from the r cyclic codes in the code chain and is a specially localized subcode of the \emph{mother code} π’ž_0 in the code chain. The minimum distance d_local of each local code of π’ž_convol is lower bounded by the minimum distance d_0 of the mother code π’ž_0 in the code chain. The local structure of π’ž_convol allows it to be decoded based on a designed parity-check matrix of the mother code π’ž_0 using a sliding window decoding scheme.
ER  -