Improvement of the Gilbert-Varshamov Bound for Linear Codes and Quantum Codes

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 $\delta=d/n<1-1/q$, we show that an $[n,k,d]_q$ linear code exists whenever $\frac{q^{k}-1}{q-1}\;<\;\frac{c_\delta \sqrt{n}\, q^{n}}{\mathrm{Vol}_q(n,d-1)}$, for positive constant $c_\delta$ depending only on $\delta$, 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 $\delta<1-1/q^2$, we obtain an improved quantum GV bound: there exists a $q$-ary quantum code $[[n,\,n-k,\,d]]$ provided that $\frac{q^{2n-k}-1}{q-1}<\frac{c_\delta \sqrt{n}\cdot q^{2n}}{\sum_{i=0}^{d-1}\binom{n}{i}(q^2-1)^i}$. In particular, our result improves the standard quantum GV bound by an $\Omega(\sqrt{n})$ multiplicative factor.