A t-design is a combinatorial structure consisting of a collection of blocks over a set of points satisfying certain properties. The existence of t-designs given a set of parameters can be reduced to finding nonnegative integer solutions to a given integer matrix equation. The matrix in this equation can be quite large, but by prescribing the automorphism group of the design, the matrix in the equation can be made more manageable so as to allow the equation to be solved via integer programming tools; this fact was developed by Kramer and Mesner. Algorithms to generate the matrix equation generally follow a simple template. In this thesis, a generic framework for generating the Kramer-Mesner matrix equation and solving it via integer programming is presented.

A t-design is a combinatorial structure consisting of a collection of blocks over a set of points satisfying certain properties. The existence of t-designs given a set of parameters can be reduced to finding nonnegative integer solutions to a given integer matrix equation. The matrix in this equation can be quite large, but by prescribing the automorphism group of the design, the matrix in the equation can be made more manageable so as to allow the equation to be solved via integer programming tools; this fact was developed by Kramer and Mesner. Algorithms to generate the matrix equation generally follow a simple template. In this thesis, a generic framework for generating the Kramer-Mesner matrix equation and solving it via integer programming is presented.