An Example of an $\mathbb{L}(n,d)$ Linear Space with more than $n^2+n+1$ Lines

Abstract

An $\mathbb{L}(n,d)$ is a linear space with constant point degree $n+1$, lines of size $n$ and $n-d$, and with $v=n^2–d$ points. Denote by $b=n^2+n+z$ the number of lines of an $\mathbb{L}(n,d)$, then $z\ge 0$ and examples are known only if $z=0,1$ [7]. The linear spaces $\mathbb{L}(n,d)$ were introduced in [7] in relation with some classification problems of finite linear spaces. In this note, starting from the symmetric configuration ${45}_7$ of Baker [1], we give an example of $\mathbb{L}(n,d)$, with $n=7,d=4$ and $z=4$.