On orthogonal arrays attaining Rao's bounds (Q1109038)

An orthogonal array, \(\text{OA}(N,m,s,t)\) is an \(m\times N\) matrix \(A\) with entries from a set \(\{0,1,\ldots,s-1\}\). It is said to be of strength \(t\) if in any \(t\times N\) submatrix of \(A\), all possible \(t\times 1\) columns occur with th e same frequency \(\lambda\). It is said to be complete if \[ \begin{alignedat}{2} N &= \binom m0(s-1)^0 +\cdots+ \binom me(s-1)^ e \qquad &&\text{if }t=2e\\ n &= \binom m0(s-1)^0 +\cdots+ \binom me(s-1)^ e + \binom{m-1}e(s-1)^{e+1} \qquad &&\text{if }t=2e+1. \end{alignedat} \] Where the right-hand-side expressions are bounds given by \textit{C. R. Rao} [J. Roy. Statist. Soc. 38, 67-78 (1947)]. The author gives possible combinations of \(N, m, s, t\) for complete orthogonal arrays with \(s=2\) and general \(t\), and with \(t=2,3\) for general \(s\). This generalizes previous work on arrays of strength 4.