On Iso-Dual MDS Codes From Elliptic Curves

TL;DR

This paper confirms a conjecture on isometry-dual MDS elliptic codes, introduces two new constructions with specific length bounds, and applies these results to develop new entanglement-assisted quantum error correcting codes.

Contribution

It provides the first confirmation of the conjecture, presents novel constructions of isometry-dual MDS codes from elliptic curves, and extends these to quantum error correction.

Findings

Confirmed the conjecture about isometry-dual MDS elliptic codes.

Constructed two new families of isometry-dual MDS codes with explicit length bounds.

Derived two new families of MDS entanglement-assisted quantum error correcting codes.

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.