arXiv subjects

Run Zheng

Publications and source records attributed to Run Zheng.

4 recordsLinked to original sources

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↗

arXiv2025

The dimension and Bose distance of some BCH codes of length $\frac{q^{m}-1}{\lambda}$

BCH codes are important error correction codes, widely utilized due to their robust algebraic structure, multi-error correcting capability, and efficient decoding algorithms. Despite their practical importance and extensive study, their parameters, including dimension, minimum distance and Bose distance, remain largely unknown in general. This paper addresses this challenge by investigating the dimension and Bose distance of BCH codes of length $(q^m - 1)/\lambda$ over the finite field $\mathbb{F}_q$, where $\lambda$ is a positive divisor of $q - 1$. Specifically, for narrow-sense BCH codes of this length with $m \geq 4$, we derive explicit formulas for their dimension for designed distance $2 \leq \delta \leq (q^{\lfloor (2m - 1)/3 \rfloor + 1} - 1)/{\lambda} + 1$. We also provide explicit formulas for their Bose distance in the range $2 \leq \delta \leq (q^{\lfloor (2m - 1)/3 \rfloor + 1} - 1)/{\lambda}$. These ranges for $\delta$ are notably larger than the previously known results for this class of BCH codes. Furthermore, we extend these findings to determine the dimension and Bose distance for certain non-narrow-sense BCH codes of the same length. Several optimal linear codes can be obtained from these BCH codes.

cs.IT↗

arXiv2025

The dimension and Bose distance of certain primitive BCH codes

BCH codes are a significant class of cyclic codes that play an important role in both theoretical research and practical applications. Their strong error-correcting abilities and efficient encoding and decoding methods make BCH codes widely applicable in various areas, including communication systems, data storage devices, and consumer electronics. Although BCH codes have been extensively studied, the parameters of BCH codes are not known in general. Let $q$ be a prime power and $m$ be a positive integer. Denote by $\mathcal{C}_{\left(q,m,\delta)\right)}$ the narrow-sense primitive BCH code with length $q^m-1$ and designed distance $\delta$. As of now, the dimensions of $\mathcal{C}_{(q,m,\delta)}$ are fully understood only for $m \leq 2$. For $m \geq 4$, the dimensions of $\mathcal{C}_{(q,m,\delta)}$ are known only for the range $2 \leq \delta \leq q^{\lfloor (m+1)/2 \rfloor +1}$ and for a limited number of special cases. In this paper, we determined the dimension and Bose distance of $\mathcal{C}_{(q,m,\delta)}$ for $m\geq 4$ and $\delta\in [2, q^{\lfloor ( 2m-1)/{3}\rfloor+1}]. $ Additionally, we have also extended our results to some primitive BCH codes that are not necessarily narrow-sense.

cs.IT↗

arXiv2023

Linear maps preserving (p,k) norms of tensor products of matrices

Let $m,n\ge 2$ be integers. Denote by $M_n$ the set of $n\times n$ complex matrices. Let $\|\cdot\|_{(p,k)}$ be the $(p,k)$ norm on $M_{mn}$ with $1\leq k\leq mn$ and $2<p<\infty$. We show that a linear map $\phi:M_{mn}\rightarrow M_{mn}$ satisfies $$\|\phi(A\otimes B)\|_{(p,k)}=\|A\otimes B\|_{(p,k)} {\rm\quad for~ all\quad}A\in M_m {\rm ~and ~}B\in M_n$$ if and only if there exist unitary matrices $U,V\in M_{mn}$ such that $$\phi(A\otimes B)=U(\varphi_1(A)\otimes \varphi_2(B))V {\rm\quad for~ all\quad}A\in M_m {\rm~ and~ }B\in M_n,$$ where $\varphi_s$ is the identity map or the transposition map $X\to X^T$ for $s=1,2$. The result is also extended to multipartite systems.

math.FA↗