发表机构
School of Mathematics and Statistics, Xi’an Jiaotong University; Research Center for Mathematics and Interdisciplinary Sciences, Shandong University; Frontiers Science Center for Nonlinear Expectations, Ministry of Education（西安交通大学数学与统计学院; 山东大学数学交叉科学研究中心; 教育部非线性期望前沿科学中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究确定了覆盖中心为乘积形式的广义覆盖码的最优渐近率，证明乘积形式约束与线性性为渐近无代价，扩展了已有结果并解决了两个开放问题，采用概率方法结合多领域工具完成证明。
AI 中文摘要
设$G_q$为大小$q\tag{≥2}$的字母表，我们确定了满足如下条件的广义覆盖码$C\tag{⊆G_q^n}$的最优渐近率：其在$G_q^{t×n}$中的覆盖中心被约束为乘积形式$C^t$。对任意固定整数$t\tag{≥1}$和任意$\rho\tag{∈[0,1]}$，我们证明：$$\n\kappa_t(\rho,q)= \n\begin{cases} \n1-H_{q^t}(\rho),&0≤\rho<1-q^{-t},\n0,&1-q^{-t}≤\rho≤1, \n\end{cases}\n$$其中$\kappa_t(\rho,q)$表示满足$t$阶覆盖半径至多为$\rho n$的码的最小渐近率$n^{-1}\log_q|C|$，$H_{q^t}$为$q^t$元熵函数。当$q$为素数幂时，我们证明在$C≤\mathbb F_q^n$的额外要求下，该公式依然成立。因此，乘积形式约束和线性性均为渐近无代价约束：所得率为大小$q^t$字母表上的普通球覆盖率。这扩展了Elimelech与Schwartz针对无线性性约束码的近期$t=2$结果，以及Cohen与Frankl针对线性码的经典$t=1$结果，从而解决了Elimelech与Schwartz提出的两个开放问题。我们的证明为概率性的，结合了信息论与概率组合学工具，包括类型方法、Janson不等式、二阶矩方法及结构化变更论证。Janson不等式与二阶矩方法的直接应用会被高度依赖的候选误差矩阵对阻碍，我们通过将误差限制为最优指数规模的平衡精确类型类来克服该阻碍，标准类型类估计结合Shearer不等式，即可得到所选行具有指定差值的误差矩阵对数量所需的界。
英文摘要
Let $G_q$ be an alphabet of size $q\geq2$. We determine the optimal asymptotic rate of generalized covering codes $C\subseteq G_q^n$, whose covering centers in $G_q^{t\times n}$ are constrained to the product form $C^t$. For every fixed integer $t\geq1$ and every $ρ\in[0,1]$, we prove that \[ κ_t(ρ,q)= \begin{cases} 1-H_{q^t}(ρ),&0\leqρ<1-q^{-t},\\ 0,&1-q^{-t}\leqρ\leq1, \end{cases} \] where $κ_t(ρ,q)$ denotes the minimum asymptotic rate $n^{-1}\log_q|C|$ among codes whose $t$-th covering radius is at most $ρn$, and $H_{q^t}$ is the $q^t$-ary entropy function. When $q$ is a prime power, we prove that the same formula holds under the additional requirement that $C\leq\mathbb F_q^n$. Thus, both the product-form constraint and linearity are asymptotically cost-free: the resulting rate is the ordinary sphere-covering rate over an alphabet of size $q^t$. This extends the recent $t=2$ result of Elimelech and Schwartz for codes without a linearity constraint and the classical $t=1$ result of Cohen and Frankl for linear codes, thereby resolving both open problems posed by Elimelech and Schwartz. Our proofs are probabilistic and combine tools from information theory and probabilistic combinatorics, including the method of types, Janson's inequality, the second-moment method, and a structured alteration argument. Direct applications of Janson's inequality and the second-moment method are obstructed by highly dependent pairs of candidate error matrices. We overcome this obstruction by restricting the errors to a balanced exact-type class of optimal exponential size. Standard type-class estimates, together with Shearer's inequality, then give the required bounds on the number of error-matrix pairs whose selected rows have a prescribed difference.
Comments18 pages
