  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.
