{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,11]],"date-time":"2026-03-11T12:17:04Z","timestamp":1773231424065,"version":"3.50.1"},"reference-count":9,"publisher":"Association for Computing Machinery (ACM)","issue":"4","license":[{"start":{"date-parts":[[2007,12,1]],"date-time":"2007-12-01T00:00:00Z","timestamp":1196467200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.acm.org\/publications\/policies\/copyright_policy#Background"}],"funder":[{"DOI":"10.13039\/100000143","name":"Division of Computing and Communication Foundations","doi-asserted-by":"publisher","award":["CCF-0514585"],"award-info":[{"award-number":["CCF-0514585"]}],"id":[{"id":"10.13039\/100000143","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["ACM Commun. Comput. Algebra"],"published-print":{"date-parts":[[2007,12]]},"abstract":"<jats:p>\n            A\n            <jats:italic>Barker sequence<\/jats:italic>\n            is a finite sequence\n            <jats:italic>a<\/jats:italic>\n            <jats:sub>o<\/jats:sub>\n            , ...,\n            <jats:italic>a<\/jats:italic>\n            <jats:sub>\n              <jats:italic>n<\/jats:italic>\n              -1\n            <\/jats:sub>\n            , each term \u00b11, for which every sum \u03a3\n            <jats:italic>\n              <jats:sub>i<\/jats:sub>\n              a\n              <jats:sub>i<\/jats:sub>\n              a\n              <jats:sub>i<\/jats:sub>\n              +k\n            <\/jats:italic>\n            with 0 &lt;\n            <jats:italic>k<\/jats:italic>\n            &lt;\n            <jats:italic>n<\/jats:italic>\n            is either 0, 1, or -- 1. It is widely conjectured that no Barker sequences of length\n            <jats:italic>n<\/jats:italic>\n            &gt; 13 exist, and this conjecture has been verified for the case when\n            <jats:italic>n<\/jats:italic>\n            is odd. We show that in this case the problem can in fact be reduced to a question of irreducibility for a certain family of univariate polynomials: No Barker sequence of length 2\n            <jats:italic>m<\/jats:italic>\n            + 1 exists if a particular integer polynomial of degree 4\n            <jats:italic>m<\/jats:italic>\n            is irreducible over Q. A proof of irreducibility for this family would thus provide a short, alternative proof that long Barker sequences of odd length do not exist. However, we also prove that the polynomials in question are always reducible modulo\n            <jats:italic>p<\/jats:italic>\n            , for every prime\n            <jats:italic>p<\/jats:italic>\n            .\n          <\/jats:p>","DOI":"10.1145\/1358183.1358185","type":"journal-article","created":{"date-parts":[[2008,4,8]],"date-time":"2008-04-08T15:40:00Z","timestamp":1207669200000},"page":"118-121","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":6,"title":["Irreducible polynomials and barker sequences"],"prefix":"10.1145","volume":"41","author":[{"given":"Peter","family":"Borwein","sequence":"first","affiliation":[{"name":"Simon Fraser University, Burnaby, B.C., Canada"}]},{"given":"Erich","family":"Kaltofen","sequence":"additional","affiliation":[{"name":"North Carolina State University, Raleigh, North Carolina"}]},{"given":"Michael J.","family":"Mossinghoff","sequence":"additional","affiliation":[{"name":"Davidson College, Davidson, North Carolina"}]}],"member":"320","published-online":{"date-parts":[[2007,12]]},"reference":[{"key":"e_1_2_1_1_1","first-page":"273","volume-title":"Communication Theory","author":"Barker R. H.","year":"1953","unstructured":"R. H. Barker , Group synchronizing of binary digital systems , Communication Theory (W. Jackson, ed.), Academic Press , New York , 1953 , pp. 273 -- 287 . R. H. Barker, Group synchronizing of binary digital systems, Communication Theory (W. Jackson, ed.), Academic Press, New York, 1953, pp. 273--287."},{"key":"e_1_2_1_2_1","series-title":"Lecture Notes in Math.","doi-asserted-by":"crossref","DOI":"10.1007\/BFb0061260","volume-title":"Cyclic Difference Sets","author":"Baumert L. D.","year":"1971","unstructured":"L. D. Baumert , Cyclic Difference Sets , Lecture Notes in Math. , vol. 182 , Springer-Verlag , Berlin , 1971 . L. D. Baumert, Cyclic Difference Sets, Lecture Notes in Math., vol. 182, Springer-Verlag, Berlin, 1971."},{"key":"e_1_2_1_3_1","volume-title":"U.K.","author":"Borwein P.","year":"2006","unstructured":"P. Borwein and M. J. Mossinghoff , Barker sequences and flat polynomials, Number Theory and Polynomials (Bristol , U.K. , 2006 ) (J. McKee and C. Smyth, eds.), London Math. Soc. Lecture Note Ser., vol. 352 , Cambridge Univ. Press , 2008. P. Borwein and M. J. Mossinghoff, Barker sequences and flat polynomials, Number Theory and Polynomials (Bristol, U.K., 2006) (J. McKee and C. Smyth, eds.), London Math. Soc. Lecture Note Ser., vol. 352, Cambridge Univ. Press, 2008."},{"key":"e_1_2_1_4_1","doi-asserted-by":"publisher","DOI":"10.1016\/0097-3165(90)90046-Y"},{"issue":"2","key":"e_1_2_1_5_1","first-page":"4","article-title":"Barker sequences and difference sets","volume":"38","author":"Eliahou S.","year":"1992","unstructured":"S. Eliahou and M. Kervaire , Barker sequences and difference sets , Enseign. Math. ( 2 ) 38 ( 1992), no. 3 -- 4 , 345--382. Corrigendum, ibid. 40 (1994), no. 1--2, 109--111. S. Eliahou and M. Kervaire, Barker sequences and difference sets, Enseign. Math. (2) 38 (1992), no. 3--4, 345--382. Corrigendum, ibid. 40 (1994), no. 1--2, 109--111.","journal-title":"Enseign. Math."},{"key":"e_1_2_1_6_1","doi-asserted-by":"publisher","DOI":"10.1007\/s10623-004-1703-7"},{"key":"e_1_2_1_7_1","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1961-0125026-2"},{"key":"e_1_2_1_8_1","first-page":"9","article-title":"Barker codes of even length","volume":"51","author":"Turyn R.","year":"1963","unstructured":"R. Turyn , On Barker codes of even length , IEEE Trans. Inform. Theory 51 ( 1963 ), no. 9 , 1256. R. Turyn, On Barker codes of even length, IEEE Trans. Inform. Theory 51 (1963), no. 9, 1256.","journal-title":"IEEE Trans. Inform. Theory"},{"key":"e_1_2_1_9_1","doi-asserted-by":"publisher","DOI":"10.2140\/pjm.1965.15.319"}],"container-title":["ACM Communications in Computer Algebra"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/1358183.1358185","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/1358183.1358185","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,18]],"date-time":"2025-06-18T13:56:04Z","timestamp":1750254964000},"score":1,"resource":{"primary":{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/1358183.1358185"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007,12]]},"references-count":9,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2007,12]]}},"alternative-id":["10.1145\/1358183.1358185"],"URL":"https:\/\/doi.org\/10.1145\/1358183.1358185","relation":{},"ISSN":["1932-2240"],"issn-type":[{"value":"1932-2240","type":"print"}],"subject":[],"published":{"date-parts":[[2007,12]]},"assertion":[{"value":"2007-12-01","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}