Mathematics – Combinatorics
Scientific paper
2005-09-15
Discrete Math. 308(15) 2008, 3322-3334
Mathematics
Combinatorics
19 pages. V2: revised, general case q=2t is added. Submitted to Discr. Math
Scientific paper
10.1016/j.disc.2007.06.038
A subset $S$ of $\{0,1,...,2t-1\}^n$ is called a $t$-fold MDS code if every line in each of $n$ base directions contains exactly $t$ elements of $S$. The adjacency graph of a $t$-fold MDS code is not connected if and only if the characteristic function of the code is the repetition-free sum of the characteristic functions of $t$-fold MDS codes of smaller lengths. In the case $t=2$, the theory has the following application. The union of two disjoint $(n,4^{n-1},2)$ MDS codes in $\{0,1,2,3\}^n$ is a double-MDS-code. If the adjacency graph of the double-MDS-code is not connected, then the double-code can be decomposed into double-MDS-codes of smaller lengths. If the graph has more than two connected components, then the MDS codes are also decomposable. The result has an interpretation as a test for reducibility of $n$-quasigroups of order 4. Keywords: MDS codes, n-quasigroups, decomposability, reducibility, frequency hypercubes, latin hypercubes
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