The Maximal Length of q-ary MDS Elliptic Codes Is Close to

Abstract

Abstract Determining the maximal length of MDS codes with certain dimension is one of the central topics in coding theory and finite geometry. The MDS Main Conjecture states that the maximal length of a non-trivial $q$-ary MDS code of dimension $k$ is $q+1$ except when $q$ is even and $k=3$ or $k=q-1$. We prove that the maximal length of non-trivial $q$-ary MDS elliptic codes is close to $\frac{q}{2}$, which gives an affirmative answer to a conjecture of Li, Wan, and Zhang. Moreover, we apply our result to derive an answer to a question on subset sums in finite abelian groups from elliptic curves.