{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/new-results-on-periodic-golay-pairs","title":"New Results on Periodic Golay Pairs","arxiv_id":"2408.15611","date":"2024-08-28","proceeding":null,"authors":["Tyler Lumsden","Ilias Kotsireas","Curtis Bright"],"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 $106$.","url_abs":"https://arxiv.org/abs/2408.15611v3","url_pdf":"https://arxiv.org/pdf/2408.15611v3.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"new-results-on-periodic-golay-pairs","repo_url":"https://github.com/tylerlumsden/GolayPair","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}