广义打包-覆盖猜想的证明

中文摘要 AI 辅助

Elimelech、Firer和Schwartz提出的广义打包-覆盖猜想断言：对于每个线性码$\mathcal{C}$和每个可容许阶$t$，第$t$个广义汉明重量$d_t(\mathcal{C})$与第$t$个广义覆盖半径$R_t(\mathcal{C})$满足$d_t(\mathcal{C})\le 2R_t(\mathcal{C})+2$。我们给出了该猜想在任意有限域上的任意线性码和任意可容许阶下的计算机辅助证明。结合该猜想的奇偶校验重构、对疑似反例长度的界以及连续删余论证，我们解决了所有阶$t\ge 32$的情况，并将剩余阶归结为有限个参数元组，通过精确的计算机验证将其排除。

英文摘要

The generalized packing--covering conjecture of Elimelech, Firer and Schwartz asserts that, for every linear code $\mathcal{C}$ and every admissible order $t$, the $t$-th generalized Hamming weight $d_t(\mathcal{C})$ and the $t$-th generalized covering radius $R_t(\mathcal{C})$ satisfy $d_t(\mathcal{C})\le 2R_t(\mathcal{C})+2$. We give a computer-assisted proof of the conjecture for every linear code over every finite field and every admissible order. Combining a parity-check reformulation of the conjecture, bounds on the length of putative counterexamples, and successive puncturing arguments, we settle all orders $t\ge 32$ and reduce the remaining orders to finitely many parameter tuples, which we exclude by an exact computer verification.

发表机构

• Université de Rennes（雷恩大学）
• University of Naples Federico II（那不勒斯费德里科二世大学）

机构由 AI 辅助整理，请以论文原文为准。