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

Chen Yuan, Ruiqi Zhu

TL;DR

This work tackles improving the Gilbert–Varshamov bound for both classical $q$-ary linear codes and quantum codes. It introduces a concise probabilistic framework based on Bonferroni inequalities to bound the chance that many random codewords fall inside a Hamming ball, first achieving a constant-factor improvement and then extending to an $\Omega(\sqrt{n})$ multiplicative gain. The quantum extension leverages symplectic self-orthogonality to translate the classical improvement into the quantum setting, yielding an improved quantum GV bound with the same $\sqrt{n}$ factor under the regime $\delta<1-1/q^{2}$. Together, these results provide stronger existence guarantees for both classical and stabilizer-based quantum codes and advance the understanding of GV-bound improvements in finite and asymptotic regimes.

Abstract

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.

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

TL;DR

This work tackles improving the Gilbert–Varshamov bound for both classical -ary linear codes and quantum codes. It introduces a concise probabilistic framework based on Bonferroni inequalities to bound the chance that many random codewords fall inside a Hamming ball, first achieving a constant-factor improvement and then extending to an multiplicative gain. The quantum extension leverages symplectic self-orthogonality to translate the classical improvement into the quantum setting, yielding an improved quantum GV bound with the same factor under the regime . Together, these results provide stronger existence guarantees for both classical and stabilizer-based quantum codes and advance the understanding of GV-bound improvements in finite and asymptotic regimes.

Abstract

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 -ary linear codes. For relative distance , we show that an linear code exists whenever , for positive constant depending only on , where denotes the volume of a -ary Hamming ball. We further adapt this approach to the quantum setting by analyzing symplectic self-orthogonal structures. For , we obtain an improved quantum GV bound: there exists a -ary quantum code provided that . In particular, our result improves the standard quantum GV bound by an multiplicative factor.
Paper Structure (11 sections, 27 theorems, 113 equations)

This paper contains 11 sections, 27 theorems, 113 equations.

Table of Contents

  1. Introduction
  2. Our Results
  3. Improvement of Hamming GV Bound.
  4. Improvement of Quantum GV Bound.
  5. Organizations
  6. Preliminaries
  7. An Initial Improvement of the Gilbert-Varshamov Bound Using Bonferroni Inequalities
  8. A General Improvement of the Gilbert–Varshamov Bound
  9. Extending to the Quantum Gilbert-Varshamov Bound
  10. Basic Form of the Quantum Gilbert-Varshamov Bound
  11. Proof of Lemma \ref{['lem:quantum-ball']}

Key Result

Lemma 1.1

Assume $d-1= \delta n$ with some constant $\delta\in (0,1-\frac{1}{q})$. Let $\vec{v}_1,\ldots \vec{v}_\ell$ be $n\geq \ell \geq 2$ random vectors distributed uniformly at random in the Hamming ball $B_q(n,\delta n)\subseteq \mathbb{F}_q^n$. Then, the probability that $\sum_{i=1}^{\ell}\vec{v}_i\in

Theorems & Definitions (38)

  • Lemma 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Lemma 2.1: Theorem 4, ashikhmin2002nonbinary
  • Lemma 2.2: Quantum Hamming Bound, calderbank1998quantum
  • Lemma 2.3: Quantum Singleton Bound, rains2002nonbinary
  • ...and 28 more