Classification of pseudo-cyclic MDS codes

Linear codes are considered. A code is characterized by the length n, the dimension k, and the minimum distance d. An [n,k,d] code over the finite field GF(q) is said to be maximum distance separable (MDS) if d=n-k+1. A. Krishna and D.V. Sarwate (1990) investigated the existence of pseudo-cyclic MDS codes over GF(q) of length n, where n divides q-1 or q+1. It is shown that the pseudo-cyclic MDS codes constructed by Krishna and Sarwate are generalised Reed-Solomon codes. Pseudo-cyclic codes are studied over GF(q), where n and q=p^h are not relatively prime. It is proven that pseudo-cyclic [n,k] MDS codes modulo ( xⁿ-a) over GF(q) exist, if and only if n= p. Furthermore, any pseudo-cyclic [p ,k] code modulo (x^p-a) over GF(q) turns out to be MDS and generalized Reed-Solomon. It is explicitly proven that some classes of pseudo-cyclic [n,k ] MDS codes over GF(q) are generalized Reed-Solomon codes. Furthermore, pseudo-cyclic [q+1,4] MDS codes over GF(q), q even, are completely classified