Elementarteiler von Inzidenzmatrizen symmetrischer Blockpläne. (Elementary divisors of incidence matrices of symmetric block designs) (Q1088991)

By a study of the integral code generated by the rows of the incidence matrix and its extension the following results are obtained: Let \(d_ 1,...,d_ V\) \((d_ 1| d_ 2,d_ 2| d_ 3,...)\) be the elementary divisors of the incidence matrix of a symmetric \((\nu,n+\lambda,\lambda)\) design. Then \(d_{\nu}=(n+\lambda)n/g.c.d.(n,\lambda)\). Moreover, if p is a prime such that \(p| n\), \(p\nmid \lambda\) and if \(x_ p\) denotes the p-part of x, then \((d_ id_{\nu +2-i})_ p=n_ p\) for \(2\leq i\leq \nu\). For projective planes it can be shown that \(d_ 1=...=d_{3n-2}=1\), hence \(d_{n^ 2-2n+5}=...=d_{n^ 2+n}=n\) and \(d_{n^ 2+n+1}=(n+1)n\). The paper also contains some results about elementary divisors of incidence matrices G satisfying the condition \(G^ tG=nI+\lambda J\).