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

2026-07-02

AI Summary

This work addresses the systematic construction of quantum error-correcting codes with enhanced Hermitian dual distance and controllable Hermitian hull dimension by introducing a novel approach based on generalized extended codes \( C(u,a) \). By analyzing the position of the vector \( u \) relative to \( C + C^{\perp_H} \) and its interaction with minimum-weight dual codewords, the authors establish, for the first time, sufficient conditions that simultaneously increase both the Hermitian hull dimension and the dual distance. They further provide a direct construction linking such codes to entanglement-assisted quantum codes. Leveraging algebraic tools including Hermitian duality theory, codeword weight analysis, and monomial equivalence, they construct 267 new entanglement-assisted qubit codes (with \( n \leq 40 \)) and 14 qutrit codes (with \( n \leq 25 \)) surpassing entries in Grassl’s table and recent literature, among which 236 qubit and 8 qutrit codes achieve strictly improved parameters.

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.

Problem

• generalized extended codes
• Hermitian dual distance
• Hermitian hull dimension
• entanglement-assisted quantum error-correcting codes
• EAQECC

Innovation

• generalized extended codes
• Hermitian hull
• Hermitian dual distance
• entanglement-assisted quantum error-correcting codes
• monomial equivalence