Permutation automorphism groups of cyclic codes I: cyclotomic association schemes

Abstract

The Berger-Charpin conjecture predicts that the permutation automorphism group of a cyclic code is generally the q-affine group, which is generated by the shift and the Frobenius multiplier on the index set. The word {\em generally} is not made precise in the literature, and the goal of this series of papers is to study when the Berger-Charpin conjecture or its variants hold. In this first part, we establish a natural connection between the theory of cyclic codes and the theory of cyclotomic association schemes (equivalently, Schur rings over cyclic groups). Using it, we show that the q-affine group is not the correct group to expect for certain code lengths. We characterize these lengths by an arithmetic condition, and show that they have density zero.

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BibTeX RIS

BibTeX

@misc{wu2026permutation,
  title = {{Permutation automorphism groups of cyclic codes I: cyclotomic association schemes}},
  author = {Yanni Wu and Ziqing Xiang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.33050},
  eprint = {2609.33050},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.33050},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Permutation automorphism groups of cyclic codes I: cyclotomic association schemes
AU  - Yanni Wu
AU  - Ziqing Xiang
PY  - 2026
DA  - 2026/09/27
PB  - arXiv
DO  - 10.48550/arxiv.2609.33050
UR  - https://arxiv.org/abs/2609.33050
AB  - The Berger-Charpin conjecture predicts that the permutation automorphism group of a cyclic code is generally the q-affine group, which is generated by the shift and the Frobenius multiplier on the index set. The word {\em generally} is not made precise in the literature, and the goal of this series of papers is to study when the Berger-Charpin conjecture or its variants hold. In this first part, we establish a natural connection between the theory of cyclic codes and the theory of cyclotomic association schemes (equivalently, Schur rings over cyclic groups). Using it, we show that the q-affine group is not the correct group to expect for certain code lengths. We characterize these lengths by an arithmetic condition, and show that they have density zero.
ER  -

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