二元本原BCH码的广义覆盖半径的渐近界

中文摘要 AI 辅助

固定整数e≥2和r≥1，本文研究二元本原e-纠错BCH码BCH(e,m)的第r个广义覆盖半径ρᵣ(BCH(e,m))。通过将覆盖问题转化为代数几何形式并结合显式Lang-Weil估计，证明对所有足够大的m，ρᵣ(BCH(e,m))≤(r+1)e−1；当e≥7时，该结果改进了Belinsky-Zabokritskiy近期的结论。此证明提供了更简洁的几何方法，尤其得出对所有足够大的m，ρ₂(BCH(e,m))=3e−1，此前仅知该值属于{3e−1,3e}。

英文摘要

Fix integers $e\ge2$ and $r\ge1$. In this paper we study the $r$-th generalized covering radius $ρ_r\left(BCH(e,m)\right)$ 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 \[ρ_r\bigl(\BCH(e,m)\bigr)\le(r+1)e-1\] for all sufficiently large $m$. For $e\ge7$, 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 \[ρ_2\bigl(BCH(e,m)\bigr)=3e-1\] for all sufficiently large $m$. Previously it was only known that \[ρ_2\bigl(\BCH(e,m)\bigr) \in \left\{3e-1,3e\right\}\] for all sufficiently large $m$.

发表机构

• The Hong Kong University of Science and Technology（香港科技大学）
• Università degli Studi della Campania “Luigi Vanvitelli”（坎帕尼亚路易吉范维特利大学）

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