On Cyclic MDS-Codes

We investigate the question when a cyclic code is maximum distance separable (MDS). For codes of (co-)dimension 3, this question is related to permutation properties of the polynomial (xb-1)/(x-1) for a certain b. Using results on these polynomials we prove that over fields of odd characteristic the only MDS cyclic codes of dimension 3 are the Reed-Solomon codes. For codes of dimension $$0(\sqrt q )$$ we prove the same result using techniques from algebraic geometry and finite geometry. Further, we exhibit a complete q-arc over the field F _q , for even q. In the last section we discuss a connection between modular representations of the general linear group over F _q and the question of whether a given cyclic code is MDS.