The \(\lambda\)-Designs with \(e_1 = 4\)

A \(d\)-design is an \(n \times n\) \((0,1)\)-matrix \(A\) satisfying \(A^t A = \lambda J + {diag}(k_1 – \lambda, \ldots, k_n – \lambda)\), where \(A^t\) is the transpose of \(A\), \(J\) is the \(n \times n\) matrix of ones, \(k_j >\lambda > 0\) (\(1 \leq j \leq n\)), and not all \(k_i\)’s are equal. Ryser [4] and Woodall [6] showed that such an \(A\) has precisely two row sums \(r_1\) and \(r_2\) (\(r_1 > r_2\)) with \(r_1 + r_2 = n + 1\). Let \(e_1\) be the number of rows of \(A\) with sum \(r_1\). It is shown that if \(e_1 = 4\), then \(\lambda = 3\).