Generalized Extended Codes with Applications in Entanglement-Assisted Qubit and Qutrit Codes

• URL: http://arxiv.org/abs/2607.02170v1
• Date: Thu, 02 Jul 2026 13:41:14 GMT
• Title: Generalized Extended Codes with Applications in Entanglement-Assisted Qubit and Qutrit Codes
• Abstract summary: We prove that any generalized extended code is monomially equivalent to the Hermitian dual of a code which is closely related to a second kind of extended code of $Cperp_rm H$.<n>We obtain explicit criteria to independently control the expected Hermitian hull dimension and Hermitian dual distance of $C(bf u,a)$.
• Score: 34.65128303335464
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We prove that any generalized extended code is monomially equivalent to the Hermitian dual of a code which is closely related to a second kind of extended code of $\C^{\perp_{\rm H}}$. Every $[n+1,k+1]_{q^2}$ linear code $\D$ with $d(\D^{\perp_{\rm H}})>1$ is monomially equivalent to the generalized extended code $\C({\bf u},a)$ of an $[n,k]_{q^2}$ linear code $\C$ for a fixed $a\in\F_{q^2}^{*}$ and some ${\bf u}\in\F_{q^2}^{n}$. We then characterize the Hermitian hull and Hermitian dual distance of $\C({\bf u},a)$ in terms of the position of ${\bf u}$ relative to $\C+\C^{\perp_{\rm H}}$ and the interaction between ${\bf u}$ and the minimum weight codewords of $\C^{\perp_{\rm H}}$, respectively. We obtain explicit criteria to independently control the expected Hermitian hull dimension and Hermitian dual distance of $\C({\bf u},a)$. In particular, several conditions for simultaneously increasing the Hermitian hull dimension and the Hermitian dual distance of $\C({\bf u},a)$ are derived. Applying these results to the Hermitian construction for EAQECCs gives us $267$ new EA qubit codes of lengths $n \leq 40$ and $14$ new EA qutrit codes of lengths $n \leq 25$ compared to the best-known codes in Grassl's code tables and the imporvements recorded in very recent works in the literature. Among the new parameter sets, we confirm improvements for $236$ qubit and $8$ qutrit codes.

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