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PreprintCC BY 4.0arXiv

Asymptotic Bounds on Generalized Covering Radii of Binary Primitive BCH Codes

  • Maosheng Xiong
  • Chi Hoi Yip
  • Ferdinando Zullo
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Abstract

Fix integers e≥2 and r≥1. In this paper we study the r-th generalized covering radius ρr(BCH(e,m)) of the binary primitive e-error-correcting BCH code BCH(e,m). By using an algebraic-geometric reformulation of the covering problem together with an explicit Lang-Weil estimate, we prove that ρr(BCH(e,m))≤(r+1)e−1 for all sufficiently large m. For e≥7, this improves a recent result of Belinsky--Zabokritskiy. Our proof gives a substantially simpler geometric approach to this upper bound. In particular it implies that ρ2(BCH(e,m))=3e−1 for all sufficiently large m. Previously it was only known that ρ2(BCH(e,m)) ∈ {3e−1,3e} for all sufficiently large m.

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Cite

BibTeXRIS
BibTeX
@misc{xiong2026asymptotic,
  title = {{Asymptotic Bounds on Generalized Covering Radii of Binary Primitive BCH Codes}},
  author = {Maosheng Xiong and Chi Hoi Yip and Ferdinando Zullo},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.30961},
  eprint = {2608.30961},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.30961},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Asymptotic Bounds on Generalized Covering Radii of Binary Primitive BCH Codes
AU  - Maosheng Xiong
AU  - Chi Hoi Yip
AU  - Ferdinando Zullo
PY  - 2026
DA  - 2026/08/31
PB  - arXiv
DO  - 10.48550/arxiv.2608.30961
UR  - https://arxiv.org/abs/2608.30961
AB  - Fix integers e≥2 and r≥1. In this paper we study the r-th generalized covering radius ρ_r(BCH(e,m)) of the binary primitive e-error-correcting BCH code BCH(e,m). By using an algebraic-geometric reformulation of the covering problem together with an explicit Lang-Weil estimate, we prove that ρ_r(BCH(e,m))≤(r+1)e−1 for all sufficiently large m. For e≥7, this improves a recent result of Belinsky--Zabokritskiy. Our proof gives a substantially simpler geometric approach to this upper bound. In particular it implies that ρ_2(BCH(e,m))=3e−1 for all sufficiently large m. Previously it was only known that ρ_2(BCH(e,m)) ∈ {3e−1,3e} for all sufficiently large m.
ER  -