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. The further investigation in algebraic geometry codes has shown that the $q$-ary Gilbert-Varshamov bound can also be improved in the case that $q$ is an odd power of a prime but not a prime with $q > 125$. However, it remains a long-standing open problem whether the $q$-ary Gilbert-Varshamov bound would be tight for those infinitely many integers $q$ which is a prime, except for Fermat primes not less than 257, and which is a generic positive integer not being a prime power. In this paper, we prove that the $q$-ary Gilbert-Varshamov bound can be improved for all but finitely many positive integers $q\geq 2$. It is shown that $ R_q(1/2) > R_\mathrm{GV}(1/2,q) $ for all integers $q > \exp(29)$. Furthermore, we show that the growth of the rate function $R_q(\delta)$ for $\delta\in(0,1)$ fixed and $q$ growing large has a nontrivial lower bound. These new lower bounds are achieved by using codes from geometry of numbers introduced by Lenstra in the 1980s.

翻译：对于任意正整数$q\geq 2$和任意实数$\delta\in(0,1)$，设$\alpha_q(n,\delta n)$表示$\mathbb{Z}_q^n$（其中$\mathbb{Z}_q=\{0,1,\dotsc,q-1\}$且$n\in\mathbb{N}$）中最小汉明距离至少为$\delta n$的子集的最大规模。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年代获得）指出：对所有正整数$q\geq 2$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$q$为素数的偶次幂且$q\geq 49$时，Tsfasman、Vladut和Zink以及Ihara在20世纪80年代首次利用代数几何码改进了q元Gilbert-Varshamov界。进一步对代数几何码的研究表明，当$q$为素数的奇次幂（但非素数本身）且$q > 125$时，q元Gilbert-Varshamov界也可被改进。然而，对于无穷多个整数$q$（包括素数情形——除不小于257的费马素数外，以及非素数幂的一般正整数），该界是否紧致仍是一个长期未决问题。本文证明：对除有限多个正整数$q\geq 2$外的所有情形，q元Gilbert-Varshamov界均可被改进。我们证明对所有整数$q > \exp(29)$，有$ R_q(1/2) > R_\mathrm{GV}(1/2,q) $。进一步，对于固定的$\delta\in(0,1)$且$q$充分大时，我们证明速率函数$R_q(\delta)$的增长具有非平凡下界。这些新下界通过使用Lenstra在20世纪80年代引入的数的几何编码得以实现。