The Handbook of Combinatorial Designs catalogs many types of combinatorial designs, together with lists of open instances for which existence has not yet been determined. We develop a constructive protocol CPro1, which uses Large Language Models (LLMs) to generate code that constructs combinatorial designs and resolves some of these open instances. The protocol starts from a definition of a particular type of design, and a verifier that reliably confirms whether a proposed design is valid. The LLM selects strategies and implements them in code, and scaffolding provides automated hyperparameter tuning and execution feedback using the verifier. Most generated code fails, but by generating many candidates, the protocol automates exploration of a variety of standard methods (e.g. simulated annealing, genetic algorithms) and experimentation with variations (e.g. cost functions) to find successful approaches. Testing on 16 different types of designs, CPro1 constructs solutions to open instances for 6 of them: Symmetric and Skew Weighing Matrices, Equidistant Permutation Arrays, Packing Arrays, Balanced Ternary Designs, and Florentine Rectangles.