{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T07:01:11Z","timestamp":1776841271660,"version":"3.51.2"},"reference-count":31,"publisher":"American Mathematical Society (AMS)","issue":"359","license":[{"start":{"date-parts":[[2026,4,15]],"date-time":"2026-04-15T00:00:00Z","timestamp":1776211200000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/501100000038","name":"Natural Sciences and Engineering Research Council of Canada","doi-asserted-by":"publisher","award":["RGPIN-2021-03089"],"award-info":[{"award-number":["RGPIN-2021-03089"]}],"id":[{"id":"10.13039\/501100000038","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    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\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"v less-than-or-equal-to 72\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>v<\/mml:mi>\n                            <mml:mo>\n                              \u2264\n                              \n                            <\/mml:mo>\n                            <mml:mn>72<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">v \\leq 72<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    using our methods, while only lengths\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"v less-than-or-equal-to 34\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>v<\/mml:mi>\n                            <mml:mo>\n                              \u2264\n                              \n                            <\/mml:mo>\n                            <mml:mn>34<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">v \\leq 34<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    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 \u00d0okovi\u0107 and Kotsireas. We also introduce and implement a new method of \u201cmulti-level\u201d compression, where sequences are uncompressed in several steps. This method allowed us to exhaustively search all lengths\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"v less-than-or-equal-to 72\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>v<\/mml:mi>\n                            <mml:mo>\n                              \u2264\n                              \n                            <\/mml:mo>\n                            <mml:mn>72<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">v \\leq 72<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    using less than 10 compute 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\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"v greater-than 100\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>v<\/mml:mi>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mn>100<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">v&gt;100<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . We demonstrate the usefulness of our algorithms by providing the first ever examples of periodic Golay pairs of length\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"v equals 90\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>v<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mn>90<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">v = 90<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . The smallest length for which the existence of periodic Golay pairs is undecided is now\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"106\">\n                        <mml:semantics>\n                          <mml:mn>106<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">106<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    .\n                  <\/p>","DOI":"10.1090\/mcom\/4096","type":"journal-article","created":{"date-parts":[[2025,3,26]],"date-time":"2025-03-26T14:21:32Z","timestamp":1742998892000},"page":"1517-1539","source":"Crossref","is-referenced-by-count":0,"title":["New results on periodic Golay pairs"],"prefix":"10.1090","volume":"95","author":[{"given":"Tyler","family":"Lumsden","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ilias","family":"Kotsireas","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Curtis","family":"Bright","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2025,4,15]]},"reference":[{"key":"1","first-page":"159","article-title":"On nonabelian McFarland difference sets","volume":"168","author":"AbuGhneim, Omar A.","year":"2004","journal-title":"Congr. 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