On the Maximal Length of MDS Elliptic Codes

2026-05-28

AI Summary

This study addresses the maximal length of elliptic curve–based maximum distance separable (MDS) codes of a given dimension over finite fields, focusing on the case of even dimension, non-square field order, and characteristic two. By integrating algebraic geometry coding theory, point counting on elliptic curves, and combinatorial analysis, the work fully resolves the impact of restricting the support set to rational points on achievable code length. When the support is confined to rational points, the maximal length of even-dimensional MDS codes is bounded above by \( q+1+\lfloor 2\sqrt{q} \rfloor -1 \); without this restriction, lengths up to \( (q+1)/2 + \sqrt{q} \) are attainable. The paper establishes a precise, unified expression for \( \mathrm{MEC}(k,q) \), thereby providing the first tight characterizations of maximal code lengths in both settings.

Abstract

The determination of the maximal length of maximum distance separable (MDS) codes arising from elliptic curves is a central problem in coding theory. For an elliptic curve $E$ over $\mathbb{F}_q$, let $\operatorname{MEC}(k,q)$ denote the maximal length of a $q$-ary MDS elliptic code of dimension $k$. It was recently shown that $\operatorname{MEC}(k,q)\le\frac{q+1}{2}+\sqrt{q}$ for $q\ge289$ and $3\le k\le(q+1-2\sqrt{q})/10$, with equality for odd $k$ when $q$ is an odd square. This paper investigates the remaining open cases, namely even dimension $k$, non-square $q$ and fields of characteristic $2$, and provides a complete resolution of the tightness question for the two natural parity regimes of $q+1+\lfloor 2\sqrt{q}\rfloor$. We prove that if the support of $G$ (used to define the code) consists of $\mathbb{F}_q$-rational points, the bound decreases to $\frac{q+1}{2}+\sqrt{q}-1$ for even $k$. Without this restriction, we construct MDS codes attaining $\frac{q+1}{2}+\sqrt{q}$ for even $k$. More generally, we establish $\operatorname{MEC}(k,q)=\frac{q+1+\lfloor2\sqrt{q}\rfloor}{2}$ when $q+1+\lfloor2\sqrt{q}\rfloor$ is even, and $\operatorname{MEC}(k,q)=\frac{q+\lfloor2\sqrt{q}\rfloor}{2}$ when it is odd.

Problem

• MDS codes
• elliptic curves
• maximal length
• finite fields
• coding theory

Innovation

• MDS codes
• elliptic curves
• maximal length
• finite fields
• coding theory