On Iso-Dual MDS Codes From Elliptic Curves
Yunlong Zhu, Chang-An Zhao
TL;DR
This work advances iso-dual MDS codes from elliptic curves by proving a maximal-length bound n ≤ |E(F_q)|/2 for such codes, and by delivering two explicit constructions over fields of even and odd characteristics, respectively. The constructions yield MDS iso-dual codes with controllable hull dimensions and, in even characteristic, equivalence to self-dual and LCD codes, enabling hull-variation analyses. These codes are then leveraged to construct MDS EAQECCs with concrete parameter families, demonstrating practical quantum-error-correcting applications. The results illuminate hull behavior and length limits for iso-dual MDS elliptic codes, while leaving open questions about hull variation in odd characteristic cases and sharper upper bounds. Overall, the paper provides new code families and useful tools for linking classical AG codes from elliptic curves to quantum error correction.
Abstract
For a linear code $C$ over a finite field, if its dual code $C^{\perp}$ is equivalent to itself, then the code $C$ is said to be {\it isometry-dual}. In this paper, we first confirm a conjecture about the isometry-dual MDS elliptic codes proposed by Han and Ren. Subsequently, two constructions of isometry-dual maximum distance separable (MDS) codes from elliptic curves are presented. The new code length $n$ satisfies $n\le\frac{q+\lfloor2\sqrt{q}\rfloor-1}{2}$ when $q$ is even and $n\le\frac{q+\lfloor2\sqrt{q}\rfloor-3}{2}$ when $q$ is odd. Additionally, we consider the hull dimension of both constructions. In the case of finite fields with even characteristics, an isometry-dual MDS code is equivalent to a self-dual MDS code and a linear complementary dual MDS code. Finally, we apply our results to entanglement-assisted quantum error correcting codes (EAQECCs) and obtain two new families of MDS EAQECCs.