Extending MDS Codes
نویسنده
چکیده
A q-ary (n,k)-MDS code, linear or not, satisfies n≤ q+ k−1. A code meeting this bound is said to have maximum length. Using purely combinatorial methods we show that an MDS code with n = q + k− 2 can be uniquely extended to a full length code if and only if q is even. This result is best possible in the sense that there is, for example, a non-extendable 4-ary (5,4)-MDS code. It may be that the proof of our result is as interesting as the result itself. We provide a simple necessary and sufficient condition (property P ) for code extendability. In future work, this condition might be suitably modified to give an extendability condition for arbitrary (shorter) MDS codes. AMS Subject Classification: 94B25, 51E21,05B15
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