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

• URL: http://arxiv.org/abs/2607.12242v1
• Date: Tue, 14 Jul 2026 00:58:42 GMT
• Title: Quantum Codes from $r$-Nearly Self-Orthogonal Linear Codes via Jordan Canonical Form over $\mathbb{F}_{q^2}$
• Abstract summary: We introduce a Jordan-canonical-form framework for constructing $q$-ary quantum stabilizer codes from arbitrary classical linear codes over $F_q2$.<n>The framework does not require the classical linear code $mathcalC$ to satisfy the dual-containing condition (i.e., self-orthogonality)<n>Applying this construction to classical codes produces several record quantum codes that improve or supplement the best-known parameters in Grassl's tables.
• Score: 5.6342405886627
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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