On existence of two symbol complete orthogonal arrays
An orthogonal array \(\text{OA}(N,m,s,t)\) consists of an \(m\times N\) matrix \(A\) with entries from an \(s\)-set such that every \(t\times N\) submatrix of \(A\) contains \(\lambda\) copies of each \(t\times 1\) column vector, and so \(N= \lambda s^ t\). An orthogonal array that achieves Rao's bound is called complete. It is known (see the second author [J. Stat. Plann. Inference 19, No. 3, 395-400 (1988; Zbl 0655.05016)]) that a complete \(\text{OA}(N,m,2,t)\) exists whenever \(m= t+1\). The authors show that a complete \(\text{OA}(N,m,2,6)\) exists if and only if \(m= 7\) or 23 and that a complete \(\text{OA}(N,m,2,7)\) exists if and only if \(m= 8\) or 24. The existence for \(m= 23\) and 24 follows from the existence of the binary (23,12) Golay code. For \(8\leq t\leq 13\), the authors have checked that for \(m\leq 10^ 9\), a complete \(\text{OA}(N,m,2,t)\) exists only for \(m= t+1\). They conjecture that for \(t\geq 4\) with \(t\neq 6\) or 7, a complete \(\text{OA}(N,m,2,t)\) exists if and only if \(m= t+1\).
scientific article; zbMATH DE number 584023