Asymptotic Bounds on Generalized Covering Radii of Binary Primitive BCH Codes
2026-08-31 • Information Theory
Information Theory
AI summaryⓘ
The authors study how well a particular type of error-correcting code called the binary primitive BCH code can cover all possible messages when allowing a certain number of errors. They focus on the r-th generalized covering radius, which roughly measures the code's error-correcting reach. Using a new geometric method and advanced counting estimates, they prove an improved upper limit on this radius for large code sizes, especially when the error-correcting capability is at least 7. Their work also determines an exact value for the 2nd generalized covering radius for large codes, refining previous results that only narrowed it down to two possibilities.
BCH codegeneralized covering radiuserror-correcting codesbinary codesalgebraic geometryLang-Weil estimatecovering problemprimitive codeupper bound
Authors
Maosheng Xiong, Chi Hoi Yip, Ferdinando Zullo
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 $m$.