  We study $C_{a, b}$ 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 $C_{a, b}$ curves can be used to construct MDS codes
and focus on some $C_{a, b}$ curves with extra automorphisms, namely
$y^3=x^4+1, y^3=x^4-x, y^3-y=x^4$. The automorphism groups of such codes are
determined in most characteristics.
