Quantum Codes from $r$-Nearly Self-Orthogonal Linear Codes via Jordan Canonical Form over $\mathbb{F}_{q^2}$

2026-07-13

AI Summary

This work addresses the limitation in traditional constructions of quantum stabilizer codes, which require classical codes to satisfy Hermitian self-orthogonality. To overcome this constraint, the authors introduce the novel notion of “$r$-nearly self-orthogonal” codes. Starting from an arbitrary classical linear code, they explicitly construct self-orthogonal codes by leveraging Jordan canonical form decomposition, analysis of Hermitian dual spaces, and rank-one perturbation techniques. They further establish a sufficient criterion that guarantees preservation of the minimum distance. The resulting $q$-ary quantum codes achieve parameters $[[n+r, 2k-n+r, \geq d]]_q$, with several concrete instances either surpassing or complementing the best-known codes listed in Grassl’s table.

Abstract

We introduce a Jordan-canonical-form framework for constructing $q$-ary quantum stabilizer codes from arbitrary classical linear codes over $\F_{q^2}$. The framework does not require the classical linear code $\mathcal{C}$ to satisfy the dual-containing condition (i.e., self-orthogonality). Given a classical code $\mathcal{C}=[n,k,d]_{q^2}$ with parity-check matrix $H$, we measure the obstruction to Hermitian self-orthogonality by the rank $r=(n-k)-\dim_{\F_{q^2}}(\mathcal{C}^{\perp_h}\cap \mathcal{C})$. The ingredient code $\mathcal{C}$ is $r$-nearly dual containing, or, equivalently, $\mathcal{C}^{\perp_h}$ is $r$-nearly self-orthogonal, by which we mean that $r=\Rank(HH^{\dagger})=\dim_{\F_{q^2}}(\mathcal{C}^{\perp_h})-\dim_{\F_{q^2}}(\mathcal{C}^{\perp_h}\cap \mathcal{C})$. By systematically reducing the rank of the Hermitian inner-product matrix $A=HH^{\dagger}$ through rank-one perturbations along the Jordan basis $W=P^{-1}$ of the decomposition $A=PJ_AP^{-1}$, we construct an explicit Hermitian self-orthogonal code $\mathcal{C}_{\mathrm{so}}=[n+r,n-k]_{q^2}$. A sufficient distance-preservation criterion guarantees that the resulting $q$-ary quantum code has parameters $[[n+r,2k-n+r,\geq d]]_q$. Applying this construction to classical codes produces several record quantum codes that improve or supplement the best-known parameters in Grassl's tables.

Problem

• quantum codes
• self-orthogonal codes
• Hermitian inner product
• classical linear codes
• stabilizer codes

Innovation

• Jordan canonical form
• r-nearly self-orthogonal codes
• quantum stabilizer codes
• Hermitian inner product
• rank-one perturbation