On MDS extensions of generalized Reed- Solomon codes

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.