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

2026-01-26

AI Summary

This work aims to surpass the long-standing Gilbert–Varshamov (GV) bound in both classical and quantum coding theory, particularly under the constraints of linearity and quantum self-orthogonality. To this end, the authors introduce a unified and streamlined probabilistic framework that analyzes the distribution of codewords over the $q$-ary space, incorporates volume estimates of Hamming balls, and integrates symplectic self-orthogonality to accommodate quantum constraints. This approach yields the first simultaneous improvement over the classical GV bound for linear codes and its quantum counterpart, overcoming the limitations of conventional techniques that struggle to adapt to quantum self-orthogonality. For classical codes with relative distance $\delta < 1 - 1/q$ and quantum codes with $\delta < 1 - 1/q^2$, the method guarantees the existence of codes with rates exceeding the standard GV bound by $\Omega(\sqrt{n})$.

Technology Category

• Machine Learning: Quantum Machine Learning
• Constraint Satisfaction and Optimization: Constraint Optimization
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

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 $\delta=d/n<1-1/q$, 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.

Problem

• Gilbert-Varshamov bound
• linear codes
• quantum codes
• error-correcting codes
• self-orthogonality

Innovation

• Gilbert-Varshamov bound
• linear codes
• quantum error-correcting codes
• probabilistic method
• symplectic self-orthogonality