On MDS extensions of generalized Reed- Solomon codes

An linear code over GF is said to be {em maximum distance separable} (MDS) if . It is shown that an generalized Reed-Solomon code such that 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 in the above range can be {em uniquely} extended to a maximal MDS code of length , and that generalized Reed-Solomon codes of length and dimension is even) do not have MDS extensions. Hence, in cases where the MDS code is essentially unique, MDS codes with do not exist.