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

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 \( z \geq 0 \) 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 \( z=4 \).