On MDS extensions of generalized Reed-Solomon codes

AbsrructAn ( n, k, d) linear code over F = GF( q) is said to be ntcrximunt dktunce separable (MDS) if d = n k + 1. It is shown that an (II, k, FI k + 1) generalized Reed-Solomon code such that 2 I k 5 n 1 ((I 1)/2] (k + 3 if q is even) can be extended by one digit while preserving the MDS property if and only if the resulting extended code is also a generalized Reed-Solomon code. It follows that a generalized Reed-Solomon code with k in the above range can be uniqueb extended to a maximal MDS code of length q + 1, and that generalized Reed-Solomon codes of length y + 1 and dimension 2 < k I 1 q/2] + 2 (X ;t 3 if q is even) do not have MDS extensions. Hence, in cases where the (q + 1, k ) MDS code is essentially unique, (n, k) MDS codes with n > q + 1 do not exist.