Asymptotic Bounds on Generalized Covering Radii of Binary Primitive BCH Codes

2026-08-31

AI Summary

本文研究了二进制原始BCH码的广义覆盖半径问题，通过代数几何方法和Lang-Weil估计，证明了对于足够大的m，r-次广义覆盖半径有一个上界，并简化了其几何证明过程。

Abstract

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 $m$.

Problem

• BCH codes
• covering radius
• asymptotic bounds
• algebraic-geometric

Innovation

• Generalized Covering Radius
• BCH Codes
• Algebraic-Geometric Reformulation
• Lang-Weil Estimate