PreprintarXiv licencearXiv
A density theorem for non-standard irreducible cyclic codes
Abstract
We prove a fixed-characteristic density form of the Berger-Charpin conjecture for non-degenerate irreducible cyclic codes: for each prime p, the lengths admitting such a non-standard code over some finite field of characteristic p have natural density zero. Thus, for almost every length coprime to p, every such code over every finite field of characteristic p has only the permutation automorphisms generated by the cyclic shift and the Frobenius map. The proof uses the Feng--Hollmann--Li--Xiang classification for non-standard irreducible cyclic codes, a normal form for the deriving constructions in that paper, and a theorem by Wiertelak on multiplicative orders, in the form recalled by Moree.
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BibTeX
@misc{feng2026density,
title = {{A density theorem for non-standard irreducible cyclic codes}},
author = {Tao Feng and Henk D. L. Hollmann and Weicong Li and Qing Xiang},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.00925},
eprint = {2611.00925},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.00925},
}
RIS
TY - UNPB
TI - A density theorem for non-standard irreducible cyclic codes
AU - Tao Feng
AU - Henk D. L. Hollmann
AU - Weicong Li
AU - Qing Xiang
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.00925
UR - https://arxiv.org/abs/2611.00925
AB - We prove a fixed-characteristic density form of the Berger-Charpin conjecture for non-degenerate irreducible cyclic codes: for each prime p, the lengths admitting such a non-standard code over some finite field of characteristic p have natural density zero. Thus, for almost every length coprime to p, every such code over every finite field of characteristic p has only the permutation automorphisms generated by the cyclic shift and the Frobenius map. The proof uses the Feng--Hollmann--Li--Xiang classification for non-standard irreducible cyclic codes, a normal form for the deriving constructions in that paper, and a theorem by Wiertelak on multiplicative orders, in the form recalled by Moree.
ER -