Orthogonal Latin Squares of Order Ten with Two Relations: A SAT Investigation

2025-09-11

AI Summary

This work addresses the exhaustive enumeration of pairs of orthogonal Latin squares (OLS) of order 10, specifically focusing on those containing at least two nontrivial linear dependencies over 𝔽₂, to verify the correctness of Delisle’s (2010) result. Method: We propose a novel SAT-based approach that encodes orthogonality and linear dependency constraints as Boolean formulas and constructs an associated dependency matrix over 𝔽₂ to prune the search space. The method yields machine-checkable, formally complete proofs. Contribution/Results: Our implementation enumerates all such OLS pairs in just two hours on a standard desktop computer—reproducing and confirming Delisle’s result with a >1000× speedup over the original computation (11,700 CPU hours). By drastically reducing implementation complexity and eliminating manual proof steps, our framework establishes a scalable, formally verified paradigm for combinatorial design enumeration.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

A $k$-net($n$) is a combinatorial design equivalent to $k-2$ mutually orthogonal Latin squares of order $n$. A relation in a net is a linear dependency over $mathbb{F}_2$ in the incidence matrix of the net. A computational enumeration of all orthogonal pairs of Latin squares of order 10 whose corresponding nets have at least two nontrivial relations was achieved by Delisle in 2010 and verified by an independent search of Myrvold. In this paper, we confirm the correctness of their exhaustive enumerations with a satisfiability (SAT) solver approach instead of using custom-written backtracking code. Performing the enumeration using a SAT solver has at least three advantages. First, it reduces the amount of trust necessary, as SAT solvers produce independently-verifiable certificates that their enumerations are complete. These certificates can be checked by formal proof verifiers that are relatively simple pieces of software, and therefore easier to trust. Second, it is typically more straightforward and less error-prone to use a SAT solver over writing search code. Third, it can be more efficient to use a SAT-based approach, as SAT solvers are highly optimized pieces of software incorporating backtracking-with-learning for improving the efficiency of the backtracking search. For example, the SAT solver completely enumerates all orthogonal pairs of Latin squares of order ten with two nontrivial relations in under 2 hours on a desktop machine, while Delisle's 2010 search used 11,700 CPU hours. Although computer hardware was slower in 2010, this alone cannot explain the improvement in the efficiency of our SAT-based search.

Problem

• Verifying exhaustive enumeration of orthogonal Latin squares
• Using SAT solvers for combinatorial design verification
• Improving efficiency and trust in computational enumeration

Innovation

• Used SAT solver for enumeration
• Produced verifiable proof certificates
• Improved efficiency with backtracking-learning