{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,21]],"date-time":"2025-10-21T15:35:53Z","timestamp":1761060953845,"version":"3.37.3"},"reference-count":32,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2020,1,9]],"date-time":"2020-01-09T00:00:00Z","timestamp":1578528000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2020,1,9]],"date-time":"2020-01-09T00:00:00Z","timestamp":1578528000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Cryptogr. Commun."],"published-print":{"date-parts":[[2020,7]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>In this work, we describe a double bordered construction of self-dual codes from group rings. We show that this construction is effective for groups of order 2<jats:italic>p<\/jats:italic> where <jats:italic>p<\/jats:italic> is odd, over the rings <jats:inline-formula><jats:alternatives><jats:tex-math>$\\mathbb {F}_{2}+u\\mathbb {F}_{2}$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mrow>\n                      <mml:mi>F<\/mml:mi>\n                    <\/mml:mrow>\n                    <mml:mrow>\n                      <mml:mn>2<\/mml:mn>\n                    <\/mml:mrow>\n                  <\/mml:msub>\n                  <mml:mo>+<\/mml:mo>\n                  <mml:mi>u<\/mml:mi>\n                  <mml:msub>\n                    <mml:mrow>\n                      <mml:mi>F<\/mml:mi>\n                    <\/mml:mrow>\n                    <mml:mrow>\n                      <mml:mn>2<\/mml:mn>\n                    <\/mml:mrow>\n                  <\/mml:msub>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:tex-math>$\\mathbb {F}_{4}+u\\mathbb {F}_{4}$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mrow>\n                      <mml:mi>F<\/mml:mi>\n                    <\/mml:mrow>\n                    <mml:mrow>\n                      <mml:mn>4<\/mml:mn>\n                    <\/mml:mrow>\n                  <\/mml:msub>\n                  <mml:mo>+<\/mml:mo>\n                  <mml:mi>u<\/mml:mi>\n                  <mml:msub>\n                    <mml:mrow>\n                      <mml:mi>F<\/mml:mi>\n                    <\/mml:mrow>\n                    <mml:mrow>\n                      <mml:mn>4<\/mml:mn>\n                    <\/mml:mrow>\n                  <\/mml:msub>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. We demonstrate the importance of this new construction by finding many new binary self-dual codes of lengths 64, 68 and 80; the new codes and their corresponding weight enumerators are listed in several tables.<\/jats:p>","DOI":"10.1007\/s12095-019-00420-3","type":"journal-article","created":{"date-parts":[[2020,1,9]],"date-time":"2020-01-09T09:04:47Z","timestamp":1578560687000},"page":"769-784","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Double bordered constructions of self-dual codes from group rings over Frobenius rings"],"prefix":"10.1007","volume":"12","author":[{"given":"Joe","family":"Gildea","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8563-2212","authenticated-orcid":false,"given":"Rhian","family":"Taylor","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Abidin","family":"Kaya","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"A.","family":"Tylyshchak","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2020,1,9]]},"reference":[{"issue":"2","key":"420_CR1","doi-asserted-by":"publisher","first-page":"235","DOI":"10.1016\/0097-3165(90)90069-9","volume":"55","author":"F Bernhardt","year":"1990","unstructured":"Bernhardt, F., Landrock, P., Manz, O.: The extended Golay codes considered as ideals. J. Combin. Theory Ser. A 55(2), 235\u2013246 (1990)","journal-title":"J. Combin. Theory Ser. A"},{"issue":"1\u20133","key":"420_CR2","doi-asserted-by":"publisher","first-page":"37","DOI":"10.1016\/S0012-365X(02)00520-4","volume":"262","author":"K Betsumiya","year":"2003","unstructured":"Betsumiya, K., Georgiou, S., Gulliver, T.A., Harada, M., Koukouvinos, C.: On self-dual codes over some prime fields. Discrete Math. 262(1\u20133), 37\u201358 (2003)","journal-title":"Discrete Math."},{"key":"420_CR3","doi-asserted-by":"publisher","first-page":"407","DOI":"10.1016\/S0019-9958(69)90497-5","volume":"15","author":"CL Chen","year":"1969","unstructured":"Chen, C.L., Peterson, W.W., Weldon, E.J.: Some results on quasi-cyclic codes. Inf. Control. 15, 407\u2013423 (1969)","journal-title":"Inf. Control."},{"issue":"6","key":"420_CR4","doi-asserted-by":"publisher","first-page":"1319","DOI":"10.1109\/18.59931","volume":"36","author":"JH Conway","year":"1990","unstructured":"Conway, J.H., Sloane, N.J.A.: A new upper bound on the minimal distance of self-dual codes. IEEE Trans. Inform. Theory 36(6), 1319\u20131333 (1990)","journal-title":"IEEE Trans. Inform. Theory"},{"key":"420_CR5","volume-title":"Circulant Matrices","author":"PJ Davis","year":"1979","unstructured":"Davis, P.J.: Circulant Matrices. Chelsea Publishing, New York (1979)"},{"key":"420_CR6","doi-asserted-by":"publisher","first-page":"194","DOI":"10.1016\/S1071-5797(02)00027-8","volume":"9","author":"G Dorfer","year":"2018","unstructured":"Dorfer, G., Maharaj, H.: Generalized AG codes and generalized duality. Finite Fields Appl. 9, 194\u2013210 (2018)","journal-title":"Finite Fields Appl."},{"key":"420_CR7","doi-asserted-by":"publisher","first-page":"32","DOI":"10.1109\/18.746770","volume":"45","author":"ST Dougherty","year":"1999","unstructured":"Dougherty, S.T., Gaborit, P., Harada, M., Sole, P.: Type II codes over $\\mathbb {F}_{2}+u\\mathbb {F}_{2}$. IEEE Trans. Inform. Theory 45, 32\u201345 (1999)","journal-title":"IEEE Trans. Inform. Theory"},{"key":"420_CR8","doi-asserted-by":"publisher","unstructured":"Dougherty, S.T., Gildea, J., Kaya, A.: Quadruple bordered constructions of self-dual codes from group rings. Cryptogr. Commun. https:\/\/doi.org\/10.1007\/s12095-019-00380-8 (2019)","DOI":"10.1007\/s12095-019-00380-8"},{"issue":"9","key":"420_CR9","doi-asserted-by":"publisher","first-page":"2115","DOI":"10.1007\/s10623-017-0440-7","volume":"86","author":"J Gildea","year":"2018","unstructured":"Dougherty S.T., Gildea, J., Taylor, R., Tylshchak, A.: Group rings, G-codes and constructions of self-dual and formally self-dual codes. Des. Codes Cryptogr. 86 (9), 2115\u20132138 (2018)","journal-title":"Des. Codes Cryptogr."},{"key":"420_CR10","doi-asserted-by":"publisher","first-page":"108","DOI":"10.1016\/j.ffa.2019.02.004","volume":"57","author":"ST Dougherty","year":"2019","unstructured":"Dougherty, S.T., Gildea, J., Korban, A., Kaya, A., Tylshchak, A., Yildiz, B.: Bordered constructions of self-dual codes from group rings and new extremal binary self-dual codes. Finite Fields Appl. 57, 108\u2013127 (2019)","journal-title":"Finite Fields Appl."},{"issue":"6","key":"420_CR11","doi-asserted-by":"publisher","first-page":"2036","DOI":"10.1109\/18.641574","volume":"43","author":"ST Dougherty","year":"1997","unstructured":"Dougherty, S.T., Harada, M., Gulliver, T.A.: Extremal binary self-dual codes. IEEE Trans. Inform. Theory 43(6), 2036\u20132047 (1997)","journal-title":"IEEE Trans. Inform. Theory"},{"key":"420_CR12","doi-asserted-by":"publisher","first-page":"14","DOI":"10.1016\/j.ffa.2009.11.004","volume":"16","author":"ST Dougherty","year":"2010","unstructured":"Dougherty, S.T., Kim, J.-L., Kulosman, H., Liu, H.: Self-dual codes over commutative Frobenius rings. Finite Fields Appl. 16, 14\u201326 (2010)","journal-title":"Finite Fields Appl."},{"issue":"3","key":"420_CR13","doi-asserted-by":"publisher","first-page":"205","DOI":"10.1016\/j.ffa.2010.11.002","volume":"17","author":"ST Dougherty","year":"2011","unstructured":"Dougherty, S.T., Yildiz, B., Karadeniz, S.: Codes over Rk, gray maps and their binary images. Finite Fields Appl. 17(3), 205\u2013219 (2011)","journal-title":"Finite Fields Appl."},{"issue":"1","key":"420_CR14","first-page":"89","volume":"6","author":"ST Dougherty","year":"2013","unstructured":"Dougherty, S.T., Yildiz, B., Karadeniz, S.: Self-dual codes over Rk and binary self-dual codes. European J. Pure Appl. Math. 6(1), 89\u2013106 (2013)","journal-title":"European J. Pure Appl. Math."},{"issue":"2","key":"420_CR15","doi-asserted-by":"publisher","first-page":"171","DOI":"10.1006\/ffta.2001.0333","volume":"8","author":"P Gaborit","year":"2002","unstructured":"Gaborit, P., Pless, V., Sole, P., Atkin, O.: Type II codes over $\\mathbb {F}_{4}$. Finite Fields Appl. 8(2), 171\u2013183 (2002)","journal-title":"Finite Fields Appl."},{"key":"420_CR16","doi-asserted-by":"publisher","first-page":"71","DOI":"10.1016\/j.ffa.2018.01.002","volume":"51","author":"J Gildea","year":"2018","unstructured":"Gildea, J., Kaya, A., Taylor, R., Yildiz, B.: Constructions for self-dual codes induced from group rings. Finite Fields Appl. 51, 71\u201392 (2018)","journal-title":"Finite Fields Appl."},{"issue":"12","key":"420_CR17","first-page":"1","volume":"324","author":"J Gildea","year":"2019","unstructured":"Gildea, J., Kaya, A., Yildiz, B.: An altered four circulant construction for self-dual codes from group rings and new extremal binary self-dual codes I. Discrete Math. 324(12), 1\u20138 (2019)","journal-title":"Discrete Math."},{"issue":"2","key":"420_CR18","doi-asserted-by":"publisher","first-page":"141","DOI":"10.1023\/A:1008276604233","volume":"11","author":"TA Gulliver","year":"1997","unstructured":"Gulliver, T.A., Harada, M.: Weight enumerators of double circulant codes and new extremal self-dual codes. Des. Codes Cryptogr. 11(2), 141\u2013150 (1997)","journal-title":"Des. Codes Cryptogr."},{"issue":"1","key":"420_CR19","doi-asserted-by":"publisher","first-page":"25","DOI":"10.1023\/A:1008206223659","volume":"11","author":"TA Gulliver","year":"1997","unstructured":"Gulliver, T.A., Harada, M.: Classification of extremal double circulant formally self-dual even codes. Des. Codes Cryptogr. 11(1), 25\u201335 (1997)","journal-title":"Des. Codes Cryptogr."},{"key":"420_CR20","first-page":"137","volume":"40","author":"TA Gulliver","year":"2008","unstructured":"Gulliver, T.A., Harada, M.: On double circulant doubly even self-dual [72, 36, 12] codes and their neighbors. Australas J. Combin. 40, 137\u2013144 (2008)","journal-title":"Australas J. Combin."},{"key":"420_CR21","doi-asserted-by":"publisher","first-page":"2064","DOI":"10.1016\/j.disc.2006.05.004","volume":"306","author":"TA Gulliver","year":"2006","unstructured":"Gulliver, T.A., Harada, M.: Classification of extremal double circulant self-dual codes of lengths 74-88. Discr. Math. 306, 2064\u20132072 (2006)","journal-title":"Discr. Math."},{"issue":"3","key":"420_CR22","first-page":"319","volume":"31","author":"T Hurley","year":"2006","unstructured":"Hurley, T.: Group rings and rings of matrices. Int. J. Pure Appl. Math. 31(3), 319\u2013335 (2006)","journal-title":"Int. J. Pure Appl. Math."},{"key":"420_CR23","doi-asserted-by":"crossref","unstructured":"Hurley, T.: Self-dual, dual-containing and related quantum codes from group rings. arXiv:0711.3983 (2007)","DOI":"10.1109\/ISIT.2007.4557511"},{"key":"420_CR24","doi-asserted-by":"publisher","first-page":"81","DOI":"10.1109\/TIT.1969.1054261","volume":"15","author":"M Karlin","year":"1969","unstructured":"Karlin, M.: New binary coding results by circulants. IEEE Trans. Inform. Theory 15, 81\u201392 (1969)","journal-title":"IEEE Trans. Inform. Theory"},{"key":"420_CR25","unstructured":"Bosma, W., Cannon, J.J., Fieker, C., Steel, A. (eds.): Handbook of Magma functions, Edition 2.16 (2010)"},{"key":"420_CR26","unstructured":"Ling, S., Sole, P.: Type II codes over $ \\mathbb {F}_{4}+u\\mathbb {F}_{4}$. Europ. J. Combinatorics 22, 983\u2013997 (2001)"},{"issue":"1","key":"420_CR27","doi-asserted-by":"publisher","first-page":"29","DOI":"10.1007\/s10623-011-9530-0","volume":"63","author":"I Mcloughlin","year":"2012","unstructured":"Mcloughlin, I.: A group ring construction of the [48, 24, 12] Type II linear block code. Des. Codes Cryptogr. 63(1), 29\u201341 (2012)","journal-title":"Des. Codes Cryptogr."},{"issue":"9","key":"420_CR28","doi-asserted-by":"publisher","first-page":"4381","DOI":"10.1109\/TIT.2008.928260","volume":"54","author":"I McLoughlin","year":"2008","unstructured":"McLoughlin, I., Hurley, T.: A group ring construction of the extended binary Golay code. IEEE Trans. Inform. Theory 54(9), 4381\u20134383 (2008)","journal-title":"IEEE Trans. Inform. Theory"},{"key":"420_CR29","doi-asserted-by":"publisher","first-page":"297","DOI":"10.1016\/j.ffa.2018.08.011","volume":"54","author":"M Shi","year":"2018","unstructured":"Shi, M., Sok, L., Sol\u00e9, P.: Self-dual codes and orthogonal matrices over large finite fields. Finite Fields and their Applications 54, 297\u2013314 (2018)","journal-title":"Finite Fields and their Applications"},{"key":"420_CR30","doi-asserted-by":"publisher","first-page":"2485","DOI":"10.1007\/s10623-017-0455-0","volume":"11","author":"M Shi","year":"2018","unstructured":"Shi, M., Qian, L., Sol\u00e9, P.: On self-dual negacirculant codes of index two and four. Designs Codes and Cryptography 11, 2485\u20132494 (2018)","journal-title":"Designs Codes and Cryptography"},{"key":"420_CR31","volume-title":"Codes and rings: theory and practice","author":"M Shi","year":"2017","unstructured":"Shi, M., Alahmadi, A., Sol\u00e9, P.: Codes and rings: theory and practice. Academic Press, New York (2017)"},{"issue":"3","key":"420_CR32","doi-asserted-by":"publisher","first-page":"635","DOI":"10.3934\/amc.2017047","volume":"11","author":"N Yankov","year":"2017","unstructured":"Yankov, N., Anev, D, Gurel, M.: Self-dual codes with an automorphism of order 13. Adv. Math. Commun. 11(3), 635\u2013645 (2017)","journal-title":"Adv. Math. Commun."}],"container-title":["Cryptography and Communications"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s12095-019-00420-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s12095-019-00420-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s12095-019-00420-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,1,8]],"date-time":"2021-01-08T00:30:17Z","timestamp":1610065817000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s12095-019-00420-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,1,9]]},"references-count":32,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2020,7]]}},"alternative-id":["420"],"URL":"https:\/\/doi.org\/10.1007\/s12095-019-00420-3","relation":{},"ISSN":["1936-2447","1936-2455"],"issn-type":[{"type":"print","value":"1936-2447"},{"type":"electronic","value":"1936-2455"}],"subject":[],"published":{"date-parts":[[2020,1,9]]},"assertion":[{"value":"3 June 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 December 2019","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"9 January 2020","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}