New Results on Periodic Golay Pairs
<|reference_start|>New Results on Periodic Golay Pairs: 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 $v \lt 100$.<|reference_end|>
@article{lumsden2024new,
    title={New Results on Periodic Golay Pairs},
    author={Tyler Lumsden, Ilias Kotsireas, Curtis Bright},
    journal={arXiv preprint arXiv:2408.15611},
    year={2024},
    archivePrefix={arXiv},
    eprint={2408.15611},
    primaryClass={cs.DM cs.SC math.CO}
}