Simplicity conditions for binary orthogonal arrays (Q2678983)

From Rao's theorem [\textit{A. S. Hedayat} et al., Orthogonal arrays. Theory and applications. New York, NY: Springer (1999; Zbl 0935.05001), Theorem 2.1], every orthogonal array of \(N\) rows, \(k\) columns, \(s\) symbols, and strength \(2u\) satisfies that \[ N\geq M(k,s,2u):=\sum_{j=0}^u\binom kj (s-1)^j. \] Let \(F(k,s,t)\) denote the minimum positive integer \(N\) such that an orthogonal array of \(N\) rows, \(k\) columns, \(s\) symbols, and strength \(t\) exists. In the case of dealing only with simple orthogonal arrays (that is, orthogonal arrays with pairwise distinct rows), this number is denoted \(F^\ast (k,s,t)\). In this paper, the authors prove that, if \(t=2u\) and \(N < 2 M(k, s, 2u)\), then the orthogonal array under consideration is simple. As a consequence, if \(t\) is even, then \(F^\ast (k,s,t)=F(k,s,t)\), whenever \(F(k,s,t)< 2 M(k, s, t)\). This result is used by the authors for determining the value \(F^\ast (k,2,t)\), for \(t\in \{2,3\}\). It implies in turn that the minimum Hamming weight of 2-CI Boolean functions is a monotone non-decreasing function when \(k\) grows. The same statement, for \(t\in\{4,5\}\), is partially dealt with.