On cyclic MDS codes of length q over GF(q) (Corresp.)
Ron M. Roth, G. Seroussi · IEEE Transactions on Information Theory · 1986
It is shown that a cyclic codeCof lengthqover GF(q)is the maximum distance separable if and only if either1) qis a prime, in which caseCis equivalent, up to a coordinate permutation, to an extended Reed-Solomon code, or2) Cis a trivial code of dimensionk \in \{1, q - 1, q \}. Hence there exists a nontrivial cyclic extended Reed-Solomon code of lengthqover GF(q)if and only ifqis a prime.