On MDS extensions of generalized Reed- Solomon codes
- 1 May 1986
- journal article
- Published by Institute of Electrical and Electronics Engineers (IEEE) in IEEE Transactions on Information Theory
- Vol. 32 (3) , 349-354
- https://doi.org/10.1109/tit.1986.1057188
Abstract
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 \neq 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 \neq 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.Keywords
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