On decomposability of 4-ary distance 2 MDS codes, double-codes, and nn-quasigroups of order 4

A subset S   of {0,1,…,2t-1}n{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=2t=2, the theory has the following application. The union of two disjoint (n,4n-1,2)(n,4n-1,2) MDS codes in {0,1,2,3}n{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.