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PreprintarXiv licencearXiv

Norm-One Torus Decompositions and Decoding of Gashkov-Sidel'nikov Codes

  • Minjia Shi
  • Shitao Li
  • Yuhong Xia
  • Tor Helleseth
  • Ferruh Ozbudak
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Abstract

Let q=3m, let K=𝔽q2, and let 𝒯={x∈ K*:NK/𝔽q(x)=1}. For both cyclic and constacyclic Gashkov-Sidel'nikov codes, we show that the set of signed parity-check column labels is precisely 𝒯. Consequently, the decoding problem separates into two stages: determining the minimum error weight associated with a syndrome S and constructing an error vector attaining this minimum. We identify the former quantity with the minimum additive length of S with respect to 𝒯 and determine it exactly by the norm and the quadratic character of 𝔽q. We also determine the complete coset-weight distribution and recover the known covering radius $3$. For the constructive part, we use quadratic-character sums and Weil bounds to construct a coset leader for every syndrome of coset weight three. The resulting procedures give complete maximum-likelihood decoders.

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Cite

BibTeXRIS
BibTeX
@misc{shi2026normone,
  title = {{Norm-One Torus Decompositions and Decoding of Gashkov-Sidel'nikov Codes}},
  author = {Minjia Shi and Shitao Li and Yuhong Xia and Tor Helleseth and Ferruh Ozbudak},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.20402},
  eprint = {2609.20402},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.20402},
}
RIS
TY  - UNPB
TI  - Norm-One Torus Decompositions and Decoding of Gashkov-Sidel'nikov Codes
AU  - Minjia Shi
AU  - Shitao Li
AU  - Yuhong Xia
AU  - Tor Helleseth
AU  - Ferruh Ozbudak
PY  - 2026
DA  - 2026/09/17
PB  - arXiv
DO  - 10.48550/arxiv.2609.20402
UR  - https://arxiv.org/abs/2609.20402
AB  - Let q=3^m, let K=𝔽_q^2, and let 𝒯={x∈ K^*:N_K/𝔽_q(x)=1}. For both cyclic and constacyclic Gashkov-Sidel'nikov codes, we show that the set of signed parity-check column labels is precisely 𝒯. Consequently, the decoding problem separates into two stages: determining the minimum error weight associated with a syndrome S and constructing an error vector attaining this minimum. We identify the former quantity with the minimum additive length of S with respect to 𝒯 and determine it exactly by the norm and the quadratic character of 𝔽_q. We also determine the complete coset-weight distribution and recover the known covering radius $3$. For the constructive part, we use quadratic-character sums and Weil bounds to construct a coset leader for every syndrome of coset weight three. The resulting procedures give complete maximum-likelihood decoders.
ER  -