Curator's Take
This article introduces a systematic way to tune the Hermitian hull and dual distance of generalized extended codes, giving researchers precise control over two key parameters that determine the performance of entanglement‑assisted quantum error‑correcting codes (EAQECCs). By exploiting these new criteria the authors construct 267 previously unknown EA qubit codes (up to length 40) and 14 EA qutrit codes (up to length 25), many of which improve on the best entries in Grassl’s tables and recent literature. The results broaden the toolbox for designing high‑rate, low‑overhead error correction that can be leveraged on near‑term hardware where shared entanglement is available, although practical deployment will still require mapping these abstract codes onto specific physical qubit architectures.
— Mark Eatherly
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 $\C^{\perp_{\rm H}}$. 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.