On MDS extensions of generalized Reed- Solomon codes

Abstract

An (n, k, d) linear code over F= GF (q) is said to be {\em maximum distance separable} (MDS) if d = n - k + 1 . It is shown that an (n, k, n - k + 1) generalized Reed-Solomon code such that 2\leq k \leq n - \lfloor (q - 1)/2 \rfloor (k \neq 3 {\rm 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 {\em uniquely} extended to a maximal MDS code of length q + 1 , and that generalized Reed-Solomon codes of length q + 1 and dimension 2\leq k \leq \lfloor q/2 \rfloor + 2 (k \neq 3 {\rm 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.