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PreprintCC BY 4.0arXiv

Counterexamples to Charpin's Conjecture on BCH codes

  • Run Zheng
  • Yaoran Yang
  • Yutong Zhang
  • Haode Yan
  • Maosheng Xiong
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Abstract

We construct an infinite family of q-ary primitive narrow-sense BCH codes whose minimum distance strictly exceeds the Bose distance; in fact, the gap between the two can be arbitrarily large as the length of the code tends to infinity. The key idea is to embed these BCH codes in a suitably large punctured generalized Reed--Muller code, whose codeword weights obey divisibility conditions supplied by Ax's theorem. This divisibility forces the minimum distance of the BCH codes far above the Bose distance. In particular, our family disproves a longstanding conjecture of Charpin asserting that this difference is at most four.

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Cite

BibTeXRIS
BibTeX
@misc{zheng2026counterexamples,
  title = {{Counterexamples to Charpin's Conjecture on BCH codes}},
  author = {Run Zheng and Yaoran Yang and Yutong Zhang and Haode Yan and Maosheng Xiong},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2607.28741},
  eprint = {2607.28741},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.28741},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Counterexamples to Charpin's Conjecture on BCH codes
AU  - Run Zheng
AU  - Yaoran Yang
AU  - Yutong Zhang
AU  - Haode Yan
AU  - Maosheng Xiong
PY  - 2026
DA  - 2026/08/28
PB  - arXiv
DO  - 10.48550/arxiv.2607.28741
UR  - https://arxiv.org/abs/2607.28741
AB  - We construct an infinite family of q-ary primitive narrow-sense BCH codes whose minimum distance strictly exceeds the Bose distance; in fact, the gap between the two can be arbitrarily large as the length of the code tends to infinity. The key idea is to embed these BCH codes in a suitably large punctured generalized Reed--Muller code, whose codeword weights obey divisibility conditions supplied by Ax's theorem. This divisibility forces the minimum distance of the BCH codes far above the Bose distance. In particular, our family disproves a longstanding conjecture of Charpin asserting that this difference is at most four.
ER  -