New Non-Equivalent (Self-Dual) MDS Codes From Elliptic Curves

It is well known that MDS codes can be constructed as algebraic geometric (AG) codes from elliptic curves. It is always interesting to construct new non-equivalent MDS codes and self-dual MDS codes. In recent years several constructions of new self-dual MDS codes from the generalized twisted Reed-Solomon codes were proposed. In this paper we construct new non-equivalent MDS and almost MDS codes from elliptic curve codes. 1) We show that there are many MDS AG codes from elliptic curves defined over ${\bf F}_q$ for any given small consecutive lengths $n$, which are not equivalent to Reed-Solomon codes and twisted Reed-Solomon codes. 2) New self-dual MDS AG codes over ${\bf F}_{{2^s}}$ from elliptic curves are constructed, which are not equivalent to Reed-Solomon codes and twisted Reed-Solomon codes. 3) Twisted versions of some elliptic curve codes are introduced such that new non-equivalent almost MDS codes are constructed. Moreover there are some non-equivalent MDS elliptic curve codes with the same length and the same dimension. The application to MDS entanglement-assisted quantum codes is given.