The concept of using basis reduction for finding \(t–(v,k,\lambda)\) designs without repeated blocks was introduced by D. L. Kreher and S. P. Radziszowski at the Seventeenth Southeastern International Conference on Combinatorics, Graph Theory and Computing. This tool and other algorithms were packaged into a system of programs that was called the design theory toolchest. It was distributed to several researchers at different institutions. This paper reports the many new open parameter situations that were settled using this toolchest.

We consider a generalization of the well-known gossip problem: Let the information of each point of a set \(X\) be conveyed to each point of a set \(Y\) by \(k\)-party conference calls. These calls are organized step-wisely, such that each point takes part in at most one call per step. During a call all the \(k\) participants exchange all the information they already know. We investigate the mutual dependence of the number \(L\) of calls and the number \(T\) of steps of such an information exchange. At first a general lower bound for \(L.k^T\) is proved. For the case that \(X\) and \(Y\) equal the set of all participants, we give lower bounds for \(L\) or \(T\), if \(T\) resp. \(L\) is as small as possible. Using these results the existence of information exchanges with minimum \(L\) and \(T\) is investigated. For \(k = 2\) we prove that for even \(n\), there is one of this kind if \(n \leq 8\).

Given a matrix in companion form over \(GF(2)\), whose characteristic polynomial is irreducible, a tridiagonal matrix, similar to the original one, is found, by constructing the similarity transformation. The theoretical basis is founded on the Lanczos tridiagonalization method, valid in the Complex domain. A variant of the Lanczos method, based on LU decomposition requirements, is modified to apply in the finite field \(GF(2)\). The work is derived from an application in VLSI design, where the matrices in companion form and in tridiagonal form represent two similar linear finite state machines, used for pseudo-random pattern generation and digital circuit testing. The construction of the similarity transformation between the matrices makes it possible to obtain directly the separate implementation of the two corresponding machines.

We determine all graphs \(G\) of order at least \(k + 1\), \(k \geq 3\), with the property that for any \(k\)-subset \(S\) of \(V(G)\) there is a unique vertex \(x, x \in V(G) – S\), which has exactly two neighbours in \(S\). Such graphs have exactly \(k + 1\) vertices and consist of a family of vertex-disjoint cycles. When \(k = 2\) it is clear that graphs with this property are the so-called friendship graphs.