Constructions of column-orthogonal strong orthogonal arrays via matchings of bipartite graphs (Q6063139)

An orthogonal array \(\mathrm{OA}(N,m,s_1\times\cdots\times s_m,t)\) is an \(N\times m\) matrix whose entries in column \(j\) are chosen from \(\{0,1,\dots,s_j-1\}\) for each \(1\le j\le m\). Its orthogonality property is that in any \(N\times t\) submatrix, all possible ordered \(t\)-tuples should appear with the same frequency. An \(N\times m\) array on \(s^2\) symbols is a strong orthogonal array SOA\((N,m,s^2,2+)\) of strength \(2+\) if any subarray of two columns can be collapsed into an \(\mathrm{OA}(N,2,s\times s^2,2)\) by applying an appropriate \(s:1\) map to the symbols in the first column or into an \(\mathrm{OA}(N,2,s^2\times s,2)\) by applying a similar map to the symbols in the second column. A related notion is that of a SOA\((N,m,s^3,2\ast)\), which has \(s^3\) symbols and is said to have strength \(2\ast\). It must satisfy a similar condition that any pair of its columns projects down to an \(\mathrm{OA}(N,2,s\times s^2,2)\) and an \(\mathrm{OA}(N,2,s^2\times s,2)\). Centering an OA involves replacing its symbols with equally spaced real numbers centered around zero. An OA has column orthogonality if its columns are orthogonal as real vectors once they have been centered. The purpose of this paper is to provide new constructions for SOAs of strength \(2+\) and \(2\ast\). Many of the SOAs are column orthogonal. For many values of the parameters, these new constructions produce SOAs with more factors (i.e. columns) than previous constructions.