The Gilbert--Varshamov (GV) bound is a central benchmark in coding theory, establishing existential guarantees for error-correcting codes and serving as a baseline for both Hamming and quantum fault-tolerant information processing. Despite decades of effort, improving the GV bound is notoriously difficult, and known improvements often rely on technically heavy arguments and do not extend naturally to the quantum setting due to additional self-orthogonality constraints. In this work we develop a concise probabilistic method that yields an improvement over the classical GV bound for $q$-ary linear codes. For relative distance $δ=d/n<1-1/q$, we show that an $[n,k,d]_q$ linear code exists whenever $\frac{q^{k}-1}{q-1}\;<\;\frac{c_δ\sqrt{n}\, q^{n}}{\mathrm{Vol}_q(n,d-1)}$, for positive constant $c_δ$ depending only on $δ$, where $\mathrm{Vol}_q(n,d-1)$ denotes the volume of a $q$-ary Hamming ball. We further adapt this approach to the quantum setting by analyzing symplectic self-orthogonal structures. For $δ<1-1/q^2$, we obtain an improved quantum GV bound: there exists a $q$-ary quantum code $[[n,\,n-k,\,d]]$ provided that $\frac{q^{2n-k}-1}{q-1}<\frac{c_δ\sqrt{n}\cdot q^{2n}}{\sum_{i=0}^{d-1}\binom{n}{i}(q^2-1)^i}$. In particular, our result improves the standard quantum GV bound by an $Ω(\sqrt{n})$ multiplicative factor.

翻译：吉尔伯特-瓦尔沙莫夫（GV）界是编码理论中的核心基准，为纠错码提供了存在性保证，并成为汉明界与量子容错信息处理的双重基线。尽管经过数十年的努力，改进GV界仍极为困难，已知的改进通常依赖于技术性极强的论证，且由于额外的自正交约束，难以自然推广至量子场景。本研究发展了一种简洁的概率方法，为$q$元线性码提供了超越经典GV界的改进结果。对于相对距离$δ=d/n<1-1/q$，我们证明当满足$\frac{q^{k}-1}{q-1}\;<\;\frac{c_δ\sqrt{n}\, q^{n}}{\mathrm{Vol}_q(n,d-1)}$时，$[n,k,d]_q$线性码必然存在，其中正常数$c_δ$仅依赖于$δ$，而$\mathrm{Vol}_q(n,d-1)$表示$q$元汉明球的体积。我们进一步将此方法适配至量子场景，通过分析辛自正交结构，对$δ<1-1/q^2$的情形得到了改进的量子GV界：当满足$\frac{q^{2n-k}-1}{q-1}<\frac{c_δ\sqrt{n}\cdot q^{2n}}{\sum_{i=0}^{d-1}\binom{n}{i}(q^2-1)^i}$时，存在$q$元量子码$[[n,\,n-k,\,d]]$。特别地，我们的结果将标准量子GV界改进了$Ω(\sqrt{n})$倍的乘性因子。