The $\lambda$-Designs with $e_1=4$

Abstract

A $d$-design is an $n\times n$ $(0,1)$-matrix $A$ satisfying $A^{t}A=\lambda J+diag(k_1–\lambda ,\dots ,k_n–\lambda )$, where $A^t$ is the transpose of $A$, $J$ is the $n\times n$ matrix of ones, $k_j>\lambda >0$ ($1\le j\le n$), 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 $\lambda =3$.