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

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 many non-Reed-Solomon type MDS codes from arbitrary genus algebraic curves can be constructed. 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
${\mathbf{F}}_{2^{s}}$
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-Reed-Solomon type MDS AG codes, which are not equivalent to known MDS twisted-Reed-Solomon codes and Roth-Lempel MDS codes, can be obtained from arbitrary genus algebraic curves. It is interesting open problem to construct explicit longer MDS AG codes from maximal curves.