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