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

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 hypercubesComment: 19 pages. V2: revised, general case q=2t is added. Submitted to Discr. Mat

Early pioneers to reversible computation

Reversible computing is one of the most intensively developing research areas nowadays. We present a survey of less known or forgotten papers to show that a transfer of ideas between different disciplines is possible

Equiangular lines in Euclidean spaces

We obtain several new results contributing to the theory of real equiangular line systems. Among other things, we present a new general lower bound on the maximum number of equiangular lines in d dimensional Euclidean space; we describe the two-graphs on 12 vertices; and we investigate Seidel matrices with exactly three distinct eigenvalues. As a result, we improve on two long-standing upper bounds regarding the maximum number of equiangular lines in dimensions d=14, and d=16. Additionally, we prove the nonexistence of certain regular graphs with four eigenvalues, and correct some tables from the literature.Comment: 24 pages, to appear in JCTA. Corrected an entry in Table