由线性码和非线性码构造的无限族3-设计
Infinite families of 3-designs from linear and nonlinear codes
- Hubei University（湖北大学）
- Hubei University of Education（湖北教育学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究有限域上线性码及其关联非线性码的码字支撑构成3-设计，并构造纠缠辅助量子纠错码和最优局部可修复码。
AI中文摘要：
编码理论与组合$t$-设计之间的联系是编码理论与组合学交叉领域的一个重要研究课题。设$q=p^m$，其中$p$为奇素数且$m\geq 2$。本文研究$\nathbb{F}_{q^2}$上的一类线性码$\mathcal{C}$及其与组合$3$-设计的联系。通过分析$\mathcal{C}$和$\mathcal{C}^{\perp}$的相关结构性质，我们证明了$\mathcal{C}$中每个非零重量的码字支撑以及$\mathcal{C}^{\perp}$中重量为$4$的码字支撑构成$3$-设计。我们进一步研究了与$\mathcal{C}$相关的一类非线性码$\mathcal{C}_2$，并证明了$\mathcal{C}_2$中每个非零汉明重量的码字支撑也构成$3$-设计。这些结果确定了更多类的线性码和非线性码，其码字支撑组合$3$-设计。特别地，非线性情形提供了支撑$3$-设计的码的额外例子，而目前关于这一主题的结果相对较少。作为应用，我们从$\mathcal{C}^{\perp}$构造了一个参数为$[[q+1,q-3,4;4]]_q$的纠缠辅助量子纠错码。我们还证明了$\mathcal{C}$是一个局部性为$3$的全符号局部可修复码。此外，我们证明了码$\mathcal{C}$满足局部可修复码的Singleton型界，因此在某些情况下是最优的。
英文摘要：
The connection between coding theory and combinatorial $t$-designs is an important research topic at the intersection of coding theory and combinatorics. Let $q=p^m$, where $p$ is an odd prime and $m\geq 2$. In this paper, we investigate a class of linear codes $\mathcal{C}$ over $\mathbb{F}_{q^2}$ and their connection with combinatorial $3$-designs. By analyzing the relevant structural properties of $\mathcal{C}$ and $\mathcal{C}^{\perp}$, we show that the supports of the codewords of every nonzero weight in $\mathcal{C}$ and {the supports of the codewords of weight $4$ in $\mathcal{C}^{\perp}$} form $3$-designs. We further investigate a class of nonlinear codes $\mathcal{C}_2$ associated with $\mathcal{C}$, and prove that the supports of the codewords of every nonzero Hamming weight in $\mathcal{C}_2$ also form $3$-designs. These results identify further classes of linear and nonlinear codes whose codewords support combinatorial $3$-designs. In particular, the nonlinear case provides additional examples of codes supporting $3$-designs, a topic for which relatively few results are currently available. As applications, we construct from $\mathcal{C}^{\perp}$ an entanglement-assisted quantum error-correcting code with parameters $[[q+1,q-3,4;4]]_q$. We also prove that $\mathcal{C}$ is an all-symbol locally repairable code with locality $3$. Furthermore, we show that the code $\mathcal{C}$ {meets the Singleton-type bound for locally repairable codes} and hence is optimal in some cases.
