Abstract
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 (x b -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.
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