Über den \(p\)-Rang von Inzidenzmatrizen. (On the \(p\)-rank of incidence matrices) (Q1082344)

Let \(r_ p\) be the \(p\)-rank of a \(b\times v\) incidence matrix G, p a prime. Assume \(G^ tG=nI+\lambda J\), \(p| n\), \(p^ 2\nmid n\) and \(p\nmid \lambda\), where I denotes the identity and J the all one matrix. Then \(v\leq 2r_ p\leq b+1\). The proof uses a special case of the following statement: Assume \(G^ tG=nI+\lambda J\), \(n=mm'\), g.c.d.\((m,m')=\) g.c.d.\((m,\lambda)=1\). Then \[ \{y\in R^ v| yG^ t=0\}\subseteq \{y| yG^ t=yJ\}\subseteq \{xG| x\in R^ b\} \] with \(R={\mathbb{Z}}/m{\mathbb{Z}}\). The proof of this statement is the same as in the paper, but uses the ring \(\{r/s| r,s\in {\mathbb{Z}}\), g.c.d.\((s,m)=1\}\).