Some New Simple \(t\)-Designs

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.