On the Maximal Length of MDS Elliptic Codes

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 $\mathrm{MEC}(k,q)$ denote the maximal length of a $q$-ary MDS elliptic code of dimension $k$. It was recently shown that $\mathrm{MEC}(k,q)\le \frac{q+1}{2}+\sqrt{q}$ for $q\ge 289$ and $3\le k\le (q+1-2\sqrt{q})\mathrm{/}10$, 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 $\mathrm{MEC}(k,q)=\frac{q+1+\lfloor 2\sqrt{q}\rfloor }{2}$ when $q+1+\lfloor 2\sqrt{q}\rfloor$ is even, and $\mathrm{MEC}(k,q)=\frac{q+\lfloor 2\sqrt{q}\rfloor }{2}$ when it is odd.