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IEEE Transactions on Information Theory

A Varshamov-Gilbert bound for a class of formally self-dual codes and related quantum codes

Vladimir D. Tonchev

Published Apr 1, 2002• 2 citations
Publisher

Abstract

It is proved that a class of q-ary (2n,n) formally self-dual codes obtained from symmetric matrices over GF (q), contains codes that meet the Varshamov-Gilbert bound. The codes are self-dual with respect to the symplectic inner product and yield quantum codes encoding one state with n q-ary qubits and having minimum distance proportional to n.

DOI
10.1109/18.992809

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It is proved that a class of q-ary (2n,n) formally self-dual codes obtained from symmetric matrices over GF (q), contains codes that meet the Varshamov-Gilbert bound. The codes are self-dual with respect to the symplectic inner product and yield quantum codes encoding one state with n q-ary qubits and having minimum distance proportional to n.
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 qqq-ary linear codes. For relative distance δ=d/n<1−1/qδ=d/n<1-1/qδ=d/n<1−1/q, we show that an [n,k,d]q[n,k,d]_q[n,k,d]q​ linear code exists whenever qk−1q−1  <  cδn qnVolq(n,d−1)\frac{q^{k}-1}{q-1}\;<\;\frac{c_δ\sqrt{n}\, q^{n}}{\mathrm{Vol}_q(n,d-1)}q−1qk−1​<Volq​(n,d−1)cδ​n​qn​, for positive constant cδc_δcδ​ depending only on δδδ, where Volq(n,d−1)\mathrm{Vol}_q(n,d-1)Volq​(n,d−1) denotes the volume of a qqq-ary Hamming ball. We further adapt this approach to the quantum setting by analyzing symplectic self-orthogonal structures. For δ<1−1/q2δ<1-1/q^2δ<1−1/q2, we obtain an improved quantum GV bound: there exists a qqq-ary quantum code [[n, n−k, d]][[n,\,n-k,\,d]][[n,n−k,d]] provided that q2n−k−1q−1<cδn⋅q2n∑i=0d−1(ni)(q2−1)i\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}q−1q2n−k−1​<∑i=0d−1​(in​)(q2−1)icδ​n​⋅q2n​. In particular, our result improves the standard quantum GV bound by an Ω(n)Ω(\sqrt{n})Ω(n​) multiplicative factor.
In this paper, a special class of two-generator quasi-twisted (QT) codes with index 2 will be presented. We explore the algebraic structure of the class of QT codes and the form of their Hermitian dual codes. A sufficient condition for self-orthogonality with Hermitian inner product is derived. Using the class of Hermitian self-orthogonal QT codes, we construct two new binary quantum codes [[70, 42, 7]]2, [[78, 30, 10]]2. According to Theorem 6 of Ref.[2], we further can get 9 new binary quantum codes. So a total of 11 new binary quantum codes are obtained and there are 10 quantum codes that can break the quantum Gilbert-Varshamov (GV) bound.