Large Language Models (LLMs) with reasoning are trained to iteratively generate and refine their answers before finalizing them, which can help with applications to mathematics and code generation. We apply code generation with reasoning LLMs to a specific task in the mathematical field of combinatorial design. This field studies diverse types of combinatorial designs, many of which have lists of open instances for which existence has not yet been determined. The Constructive Protocol CPro1 uses LLMs to generate search heuristics that have the potential to construct solutions to small open instances. Starting with a textual definition and a validity verifier for a particular type of design, CPro1 guides LLMs to select and implement strategies, while providing automated hyperparameter tuning and execution feedback. CPro1 with reasoning LLMs successfully solves long-standing open instances for 7 of 16 combinatorial design problems selected from the 2006 Handbook of Combinatorial Designs, including new solved instances for 3 of these (Bhaskar Rao Designs, Symmetric Weighing Matrices, Balanced Ternary Designs) that were unsolved by CPro1 with non-reasoning LLMs. It also solves open instances for several problems from recent (2025) literature, generating new Covering Sequences, Johnson Clique Covers, Deletion Codes, and a Uniform Nested Steiner Quadruple System.

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.