Many Non-Reed-Solomon Type MDS Codes From Arbitrary Genus Algebraic Curves

TL;DR

This paper demonstrates the existence of many non-Reed-Solomon type MDS codes derived from arbitrary genus algebraic curves, including new constructions from elliptic curves, expanding the diversity of known MDS codes.

Contribution

It proves that non-Reed-Solomon MDS codes can be constructed from any genus algebraic curve and provides explicit examples from elliptic curves, including self-dual codes.

Findings

MDS codes from higher genus curves are not equivalent to those from lower genus curves.

Constructed small-length MDS codes from elliptic curves.

Produced new self-dual MDS codes over finite fields.

Abstract

It is always interesting and important to construct non-Reed-Solomon type MDS codes in coding theory and finite geometries. In this paper, we prove that there are non-Reed-Solomon type MDS codes from arbitrary genus algebraic curves. It is proved that MDS algebraic geometry (AG) codes from higher genus curves are not equivalent to MDS AG codes from lower genus curves. For genus one case, we construct MDS AG codes of small consecutive lengths from elliptic curves. New self-dual MDS AG codes over ${\bf F}_{{2^s}}$ from elliptic curves are also constructed. These MDS AG codes are not equivalent to Reed-Solomon codes, not equivalent to known MDS twisted Reed-Solomon codes and not equivalent to Roth-Lempel MDS codes. Hence many non-equivalent MDS AG codes, which are not equivalent to Reed-Solomon codes and known MDS twisted-Reed-Solomon codes, can be obtained from arbitrary genus algebraic curves. It is interesting open problem to construct explicit longer MDS AG codes from maximal curves.