# Complex Golay Pairs up to Length 28: A Search via Computer Algebra and Programmatic SAT

## TL;DR

This paper develops a novel algorithm combining computer algebra and SAT solving to search for complex Golay pairs up to length 28, confirming their existence at lengths 24 and 26 and non-existence at others.

## Contribution

The authors introduce an improved SAT+CAS algorithm for exhaustive search of complex Golay pairs, extending the maximum length from 25 to 28.

## Key findings

- Golay pairs exist at lengths 24 and 26
- Golay pairs do not exist at lengths 23, 25, 27, and 28
- The algorithm confirms a 2002 conjecture about length 23

## Abstract

We use techniques from the fields of computer algebra and satisfiability checking to develop a new algorithm to search for complex Golay pairs. We implement this algorithm and use it to perform a complete search for complex Golay pairs of lengths up to 28. In doing so, we find that complex Golay pairs exist in the lengths 24 and 26 but do not exist in the lengths 23, 25, 27, and 28. This independently verifies work done by F. Fiedler in 2013 and confirms the 2002 conjecture of Craigen, Holzmann, and Kharaghani that complex Golay pairs of length 23 don't exist. Our algorithm is based on the recently proposed SAT+CAS paradigm of combining SAT solvers with computer algebra systems to efficiently search large spaces specified by both algebraic and logical constraints. The algorithm has two stages: first, a fine-tuned computer program uses functionality from computer algebra systems and numerical libraries to construct a list containing every sequence which could appear as the first sequence in a complex Golay pair up to equivalence. Second, a programmatic SAT solver constructs every sequence (if any) that pair off with the sequences constructed in the first stage to form a complex Golay pair. This extends work originally presented at the International Symposium on Symbolic and Algebraic Computation (ISSAC) in 2018; we discuss and implement several improvements to our algorithm that enabled us to improve the efficiency of the search and increase the maximum length we search from length 25 to 28.