On the \(p\)-rank of incidence matrices and a bound of Bruen and Ott (Q1203952)

The authors discuss the following problem: what is a lower bound of the rank \(rk_ p(B)\) of the incidence matrix \(B\) of a finite partial linear space (PLS) viewed as a matrix over a field with characteristic \(p\). The first bound they obtain results in investigating the flag matrix \(F\) of a given PLS. The authors prove that for a PLS with \(\nu\) points, \(b\) lines and \(f\) flags, and degree of any point and line congruent to 1 modulo \(p\) we have \((rk_ p(B)-1)^ 2 \geq f - \nu - b + 1\). For projective planes of order \(n\) with \(p\) dividing \(n\), this bound coincides with the bound obtained by Bruen and Ott. To obtain the second bound the authors use techniques of incidence graphs. Then they prove that for a connected PLS with, \(\nu\), \(b\) and \(f\) as above it holds \(rk_ p(B)^ 2 \geq f - \nu - b + 1\). At the end of the note the authors give explicit computer results for \(rk_ p(B)\) and \(rk_ p(F)\) of some projective planes of small order, with small \(p\).