On MDS extensions of generalized Reed- Solomon codes
G. Seroussi, Ron M. Roth · IEEE Transactions on Information Theory · 1986
An(n, k, d)linear code overF=GF(q)is said to be {\em maximum distance separable} (MDS) ifd = n - k + 1. It is shown that an(n, k, n - k + 1)generalized Reed-Solomon code such that2\leq k \leq n - \lfloor (q - 1)/2 \rfloor (k eq 3 {\rm if} qis 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 withkin the above range can be {\em uniquely} extended to a maximal MDS code of lengthq + 1, and that generalized Reed-Solomon codes of lengthq + 1and dimension2\leq k \leq \lfloor q/2 \rfloor + 2 (k eq 3 {\rm if} qis even) do not have MDS extensions. Hence, in cases where the(q + 1, k)MDS code is essentially unique,(n, k)MDS codes withn > q + 1do not exist.