The q-ary Gilbert-Varshamov bound can be improved for all but finitely many positive integers q
Xue-Bin Liang
TL;DR
This work proves that the classical q-ary Gilbert–Varshamov bound can be improved for all but finitely many positive integers q. It introduces Lenstra codes from geometry of numbers and couples them with infinite towers of Hilbert class fields, using S_c-Hilbert class field towers and adèle/idèle tools to obtain explicit lower bounds on the rate function R_q(δ). The authors establish a concrete bound R_q(δ) > 1 − δ − C (log log q)/q^{1/6} and show R_q(1/2) > R_{GV}(1/2,q) for q > exp(29), with η(δ) bounded below by 1/6, demonstrating algebraic growth in q. These results are unconditional and rely on number-field constructions rather than GRH, highlighting a new pathway to surpass the GV bound in the q-ary setting and offering implications for asymptotic code rates. The approach blends Lenstra codes, towers of Hilbert class fields, and explicit prime-distribution analyses to yield robust, quantitative improvements over the longstanding GV bound.
Abstract
For any positive integer $q\geq 2$ and any real number $δ\in(0,1)$, let $α_q(n,δn)$ denote the maximum size of a subset of $\mathbb{Z}_q^n$ with minimum Hamming distance at least $δn$, where $\mathbb{Z}_q=\{0,1,\dotsc,q-1\}$ and $n\in\mathbb{N}$. The asymptotic rate function is defined by $ R_q(δ) = \limsup_{n\rightarrow\infty}\frac{1}{n}\log_qα_q(n,δn).$ The famous $q$-ary asymptotic Gilbert-Varshamov bound, obtained in the 1950s, states that \[ R_q(δ) \geq 1 - δ\log_q(q-1)-δ\log_q\frac{1}δ-(1-δ)\log_q\frac{1}{1-δ} \stackrel{\mathrm{def}}{=}R_\mathrm{GV}(δ,q) \] for all positive integers $q\geq 2$ and $0<δ<1-q^{-1}$. 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 $R_\mathrm{GV}(δ,q)$ at a specific tangent point $δ=δ_0\in (0,1)$ of the curve $R_\mathrm{GV}(δ,q)$ for each given integer $q\geq 46$. However, the $q$-ary Gilbert-Varshamov bound $R_\mathrm{GV}(δ,q)$ at $δ=1/2$, 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 $R_\mathrm{GV}(δ,q)$ with $δ\in(0,1)$ can be improved for all but finitely many positive integers $q$. It is shown that the growth defined by $η(δ)= \liminf_{q\rightarrow\infty}\frac{1}{\log q}\log[1-δ-R_q(δ)]^{-1}$ for every $δ\in(0,1)$ has actually a nontrivial lower bound.