Pseudocyclic maximum-distance-separable codes

The (n, k) pseudocyclic maximum-distance-separable (MDS) codes modulo (xⁿ- a) over GF(q) are considered. Suppose that n is a divisor of q+1. If n is odd, pseudocyclic MDS codes exist for all k. However, if n is even, nontrivial pseudocyclic MDS codes exist for odd k (but not for even k) if a is a quadratic residue in GF(q), and they exist for even k (but not for odd k) if a is not a quadratic residue in GF(q). Also considered is the case when n is a divisor of q-1, and it is shown that pseudocyclic MDS codes exist if and only if the multiplicative order of a divides (q-1)/n, and that when this condition is satisfied, such codes exist for all k. If the condition is not satisfied, every pseudocyclic code of length n is the result of interleaving a shorter pseudocyclic code