Abstract
Determining the maximal length of MDS codes with certain dimension has been an interesting research topic in coding theory. The objective of this paper is to derive an upper bound for the maximal length of MDS elliptic codes over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb {F}_{q}$ </tex-math></inline-formula> with dimension <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$3\leq k\leq \frac {q+1-2\sqrt {q}}{10}$ </tex-math></inline-formula> . For such a range of dimension <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> , our result improves an earlier bound of Munuera and gives an affirmative solution to the conjecture of Li, Wan, and Zhang. Most notably, the proposed upper bound is tight for odd <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> in the sense that it can be achieved by some well-designed MDS elliptic codes.
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