We study Ca,b curves and their applications to coding theory.
Recently, Joyner and Ksir have suggested a decoding algorithm based on the
automorphisms of the code. We show how Ca,b curves can be used to construct
MDS codes and focus on some Ca,b curves with extra automorphisms, namely
y ³ = x⁴ + 1, y ³ = x⁴ −x, y ³ −y = x⁴. The automorphism groups of such codes
are determined in most characteristics.