Asymptotic Bounds on Generalized Covering Radii of Binary Primitive BCH Codes

Fix integers $e\ge 2$ and $r\ge 1$. In this paper we study the $r$-th generalized covering radius ${\rho }_r(BCH(e,m))$ 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 $${\rho }_r(\text{BCH}(e,m))\le (r+1)e-1$$ for all sufficiently large $m$. For $e\ge 7$, 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 $${\rho }_2(BCH(e,m))=3e-1$$ for all sufficiently large $m$. Previously it was only known that $${\rho }_2(\text{BCH}(e,m))\in \{3e-1,3e\}$$ for all sufficiently large $m$.