For any positive integer $q\geq 2$ and any real number $\delta\in(0,1)$, let $\alpha_q(n,\delta n)$ denote the maximum size of a subset of $\mathbb{Z}_q^n$ with minimum Hamming distance at least $\delta n$, where $\mathbb{Z}_q=\{0,1,\dotsc,q-1\}$ and $n\in\mathbb{N}$. The asymptotic rate function is defined by $ R_q(\delta) = \limsup_{n\rightarrow\infty}\frac{1}{n}\log_q\alpha_q(n,\delta n).$ The famous $q$-ary asymptotic Gilbert-Varshamov bound, obtained in the 1950s, states that \[ R_q(\delta) \geq 1 - \delta\log_q(q-1)-\delta\log_q\frac{1}{\delta}-(1-\delta)\log_q\frac{1}{1-\delta} \stackrel{\mathrm{def}}{=}R_\mathrm{GV}(\delta,q) \] for all positive integers $q\geq 2$ and $0<\delta<1-q^{-1}$. In the case that $q$ is an even power of a prime with $q\geq 49$, the $q$-ary Gilbert-Varshamov bound was firstly improved by using algebraic geometry codes in the works of Tsfasman, Vladut, and Zink and of Ihara in the 1980s. These algebraic geometry codes have been modified to improve the $q$-ary Gilbert-Varshamov bound $R_\mathrm{GV}(\delta,q)$ at a specific tangent point $\delta=\delta_0\in (0,1)$ of the curve $R_\mathrm{GV}(\delta,q)$ for each given integer $q\geq 46$. However, the $q$-ary Gilbert-Varshamov bound $R_\mathrm{GV}(\delta,q)$ at $\delta=1/2$, i.e., $R_\mathrm{GV}(1/2,q)$, remains the largest known lower bound of $R_q(1/2)$ for infinitely many positive integers $q$ which is a generic prime and which is a generic non-prime-power integer. In this paper, by using codes from geometry of numbers introduced by Lenstra in the 1980s, we prove that the $q$-ary Gilbert-Varshamov bound $R_\mathrm{GV}(\delta,q)$ with $\delta\in(0,1)$ can be improved for all but finitely many positive integers $q$. It is shown that the growth defined by $\eta(\delta)= \liminf_{q\rightarrow\infty}\frac{1}{\log q}\log[1-\delta-R_q(\delta)]^{-1}$ for every $\delta\in(0,1)$ has actually a nontrivial lower bound.

翻译：对于任意正整数$q\geq 2$和任意实数$\delta\in(0,1)$，令$\alpha_q(n,\delta n)$表示$\mathbb{Z}_q^n$（其中$\mathbb{Z}_q=\{0,1,\dotsc,q-1\}$且$n\in\mathbb{N}$）中满足最小汉明距离至少为$\delta n$的子集的最大尺寸。渐近速率函数定义为$ R_q(\delta) = \limsup_{n\rightarrow\infty}\frac{1}{n}\log_q\alpha_q(n,\delta n).$ 著名的q元渐近Gilbert-Varshamov界（于20世纪50年代提出）指出：对于所有正整数$q\geq 2$和$0<\delta<1-q^{-1}$，有\[ R_q(\delta) \geq 1 - \delta\log_q(q-1)-\delta\log_q\frac{1}{\delta}-(1-\delta)\log_q\frac{1}{1-\delta} \stackrel{\mathrm{def}}{=}R_\mathrm{GV}(\delta,q). \] 当q为素数偶次幂且$q\geq 49$时，Tsfasman、Vladut与Zink以及Ihara在20世纪80年代的工作首次利用代数几何码改进了q元Gilbert-Varshamov界。这些代数几何码经修正后，可在曲线$R_\mathrm{GV}(\delta,q)$的特定切点$\delta=\delta_0\in (0,1)$处改进每个给定整数$q\geq 46$对应的q元Gilbert-Varshamov界$R_\mathrm{GV}(\delta,q)$。然而，对于无穷多个正整数q（包括一般素数及一般非素数幂整数），在$\delta=1/2$处的q元Gilbert-Varshamov界$R_\mathrm{GV}(1/2,q)$仍是$R_q(1/2)$最大的已知下界。本文利用Lenstra于20世纪80年代引入的数的几何码，证明了对于除有限多个正整数q外的所有q，$\delta\in(0,1)$对应的q元Gilbert-Varshamov界$R_\mathrm{GV}(\delta,q)$均可得到改进。研究表明，对于每个$\delta\in(0,1)$，由$\eta(\delta)= \liminf_{q\rightarrow\infty}\frac{1}{\log q}\log[1-\delta-R_q(\delta)]^{-1}$定义的增长率实际上具有非平凡下界。