Abstract
The union of e disjoint MDS (or perfect) codes with distance 2 (respectively, 3) is always an e-fold MDS (perfect) code. The converse is shown to be incorrect. Moreover, if k is a multiple of 4 and n + 1 ≥ 16 is a power of two, then a k/2-fold k-ary MDS code of length m ≥ 3 and an (n + 1)/8-fold perfect code of length n exist from which no MDS (perfect) code can be isolated.
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