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Classification of pseudo-cyclic MDS codes

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2 Author(s)
Pedersen, J.P. ; Tech. Univ. of Denmark, Lyngby, Denmark ; Dahl, C.

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=ph 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 (xp-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

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Information Theory, IEEE Transactions on  (Volume:37 ,  Issue: 2 )