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We extend the idea of <jats:italic>G<\/jats:italic>-codes to composite <jats:italic>G<\/jats:italic>-codes. We show that these codes are ideals in a group ring, where the ring is a finite commutative Frobenius ring and <jats:italic>G<\/jats:italic> is an arbitrary finite group. We prove that the dual of a composite <jats:italic>G<\/jats:italic>-code is also a composite <jats:italic>G<\/jats:italic>-code. We also define quasi-composite <jats:italic>G<\/jats:italic>-codes. Additionally, we study generator matrices, which consist of the identity matrices and the composite matrices. Together with the generator matrices, the well known extension method, the neighbour method and its generalization, we find extremal binary self-dual codes of length 68 with new weight enumerators for the rare parameters <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\gamma =7,8$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>\u03b3<\/mml:mi>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:mn>7<\/mml:mn>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:mn>8<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and 9. In particular, we find 49 new such codes. Moreover, we show that the codes we find are inaccessible from other construction<\/jats:p>","DOI":"10.1007\/s10623-021-00882-8","type":"journal-article","created":{"date-parts":[[2021,5,19]],"date-time":"2021-05-19T16:06:06Z","timestamp":1621440366000},"page":"1615-1638","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Composite matrices from group rings, composite G-codes and constructions of self-dual codes"],"prefix":"10.1007","volume":"89","author":[{"given":"Steven T.","family":"Dougherty","sequence":"first","affiliation":[]},{"given":"Joe","family":"Gildea","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5206-6480","authenticated-orcid":false,"given":"Adrian","family":"Korban","sequence":"additional","affiliation":[]},{"given":"Abidin","family":"Kaya","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,5,19]]},"reference":[{"key":"882_CR1","unstructured":"Borello M., Wolfgang W.: On the Algebraic Structure of Quasi Group Codes. arXiv:1912.09167."},{"key":"882_CR2","doi-asserted-by":"publisher","first-page":"235","DOI":"10.1006\/jsco.1996.0125","volume":"24","author":"W Bosma","year":"1997","unstructured":"Bosma W., Cannon J., Playoust C.: The Magma algebra system. 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