arXiv subjects

Maosheng Xiong

Publications and source records attributed to Maosheng Xiong.

3 recordsLinked to original sources

arXiv2026

Asymptotic Bounds on Generalized Covering Radii of Binary Primitive BCH Codes

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 $m$.

cs.IT↗

arXiv2026

The generalized covering radii of Melas codes

The generalized covering radii have recently emerged as fundamental parameters of linear codes with applications to database linear querying. In this paper, we study the generalized covering radii $ρ_t(M(m,q))$ of Melas codes $M(m,q)$ over any finite field $\mathbb{F}_q$. We determine $ρ_2(M(m,q))$ for all $q$, and for a general $t \ge 3$, we prove that $ρ_t(M(m,q)) \in \left\{2t,2t+1\right\}$ for $q \in \{2,3\}$ and $ρ_t(M(m,q))=2t$ for $q \ge 4$ whenever $m$ is sufficiently large. These results extend recent work on the covering radius of Melas codes.

cs.IT↗

arXiv2026

Counterexamples to Charpin's Conjecture on BCH codes

We construct an infinite family of $q$-ary primitive narrow-sense BCH codes whose minimum distance strictly exceeds the Bose distance; in fact, the gap between the two can be arbitrarily large as the length of the code tends to infinity. The key idea is to embed these BCH codes in a suitably large punctured generalized Reed--Muller code, whose codeword weights obey divisibility conditions supplied by Ax's theorem. This divisibility forces the minimum distance of the BCH codes far above the Bose distance. In particular, our family disproves a longstanding conjecture of Charpin asserting that this difference is at most four.

cs.IT↗