On iso-dual MDS codes from elliptic curves

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 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 $n\le \frac{q+\lfloor 2\sqrt{q}\rfloor -1}{2}$ when q is even and $n\le \frac{q+\lfloor 2\sqrt{q}\rfloor -3}{2}$ 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.