Dual systems of integer vectors and their applications to the theory of \((0,1)\)-matrices (Q1181659)

The authors give a detailed account of a portion of the results described in [Math. USSR, Sb. 74, 541-554 (1993); translation from Mat. Sb. 182, 1796-1821 (1991; Zbl 0751.52007) and the preceding review (Zbl 0833.11028)]. Section 1 serves as a brief reminder of the elementary ideas and theorems in the theory of dual systems of integer \(n\)-dimensional vectors. We then consider in detail the notion of an invariant matrix, and an algorithm for enumerating a special type of dual system. With this algorithm all dual systems can be found when \(n\leq 4\), and, with one exception, all when \(n=5\). We apply this algorithm, including all details, to the case \(n=3\); in addition, we give a brief treatment of the case \(n=4\), giving only the important intermediate steps and final results. As an application of the theory of dual systems, we consider the question of whether a \((0,1)\)-matrix can be factored.