Self-Dual Near MDS Codes from Elliptic Curves

Abstract

In recent years, self-dual MDS codes have attracted a lot of attention due to theoretical interest and practical importance. Similar to self-dual MDS codes, self-dual near MDS (NMDS for short) codes have nice structures as well. From both theoretical and practical points of view, it is natural to study self-dual NMDS codes. Although there has been lots of work on NMDS codes in literature, little is known for self-dual NMDS codes. It seems more challenging to construct self-dual NMDS codes than self-dual MDS codes. The only work on construction of self-dual NMDS codes shows existence of <inline-formula> <tex-math notation= LaTeX >$q$ </tex-math></inline-formula>-ary self-dual NMDS codes of length <inline-formula> <tex-math notation= LaTeX >$q-1$ </tex-math></inline-formula> for odd prime power <inline-formula> <tex-math notation= LaTeX >$q$ </tex-math></inline-formula> or length up to 16 for some small primes <inline-formula> <tex-math notation= LaTeX >$q$ </tex-math></inline-formula> with <inline-formula> <tex-math notation= LaTeX >$q\\le 197$ </tex-math></inline-formula>. In this paper, we make use of properties of elliptic curves to construct self-dual NMDS codes. It turns out that, as long as <inline-formula> <tex-math notation= LaTeX >$2|q$ </tex-math></inline-formula> and <inline-formula> <tex-math notation= LaTeX >$n$ </tex-math></inline-formula> is even with <inline-formula> <tex-math notation= LaTeX >$4\\le n\\le q+\\lfloor 2\\sqrt {q}\\rfloor -2$ </tex-math></inline-formula>, one can construct a self-dual NMDS code of length <inline-formula> <tex-math notation= LaTeX >$n$ </tex-math></inline-formula> over <inline-formula> <tex-math notation= LaTeX >$ {\\mathbb {F}}_{q}$ </tex-math></inline-formula>. Furthermore, for odd prime power <inline-formula> <tex-math notation= LaTeX >$q$ </tex-math></inline-formula>, there exists a self-dual NMDS code of length <inline-formula> <tex-math notation= LaTeX >$n$ </tex-math></inline-formula> over <inline-formula> <tex-math notation= LaTeX >$ {\\mathbb {F}}_{q}$ </tex-math></inline-formula> if <inline-formula> <tex-math notation= LaTeX >$q\\ge 4^{n+3}\\times (n+3)^{2}$ </tex-math></inline-formula>.