New Results on Periodic Golay Pairs

TL;DR

This paper develops advanced algorithmic methods to exhaustively search for periodic Golay pairs of lengths up to 72, introduces multi-level compression techniques, and provides the first examples of such pairs at length 90, expanding the known existence range.

Contribution

It introduces new sequence generation and compression methods, enabling exhaustive searches for periodic Golay pairs up to length 72 and discovering the first pairs at length 90.

Findings

Exhaustive search for lengths up to 72 completed.

First known periodic Golay pairs of length 90 found.

Conjecture on the structure of Golay pairs supported in many lengths.

Abstract

In this paper, we provide algorithmic methods for conducting exhaustive searches for periodic Golay pairs. Our methods enumerate several lengths beyond the currently known state-of-the-art available searches: we conducted exhaustive searches for periodic Golay pairs of all lengths $v \leq 72$ using our methods, while only lengths $v \leq 34$ had previously been exhaustively enumerated. Our methods are applicable to periodic complementary sequences in general. We utilize sequence compression, a method of sequence generation derived in 2013 by Djokovi\'c and Kotsireas. We also introduce and implement a new method of "multi-level" compression, where sequences are uncompressed in several steps. This method allowed us to exhaustively search all lengths $v \leq 72$ using less than 10 CPU years. For cases of complementary sequences where uncompression is not possible, we introduce some new methods of sequence generation inspired by the isomorph-free exhaustive generation algorithm of orderly generation. Finally, we pose a conjecture regarding the structure of periodic Golay pairs and prove it holds in many lengths, including all lengths $v \lt 100$. We demonstrate the usefulness of our algorithms by providing the first ever examples of periodic Golay pairs of length $v = 90$. The smallest length for which the existence of periodic Golay pairs is undecided is now $106$.