{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T12:59:46Z","timestamp":1772283586925,"version":"3.50.1"},"reference-count":11,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2019,3,7]],"date-time":"2019-03-07T00:00:00Z","timestamp":1551916800000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["AAECC"],"published-print":{"date-parts":[[2019,6]]},"DOI":"10.1007\/s00200-019-00384-0","type":"journal-article","created":{"date-parts":[[2019,3,7]],"date-time":"2019-03-07T09:49:51Z","timestamp":1551952191000},"page":"193-206","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":13,"title":["On codes over Frobenius rings: generating characters, MacWilliams identities and generator matrices"],"prefix":"10.1007","volume":"30","author":[{"given":"Steven T.","family":"Dougherty","sequence":"first","affiliation":[]},{"given":"Eseng\u00fcl","family":"Salt\u00fcrk","sequence":"additional","affiliation":[]},{"given":"Steve","family":"Szabo","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2019,3,7]]},"reference":[{"issue":"5","key":"384_CR1","doi-asserted-by":"publisher","first-page":"355","DOI":"10.1007\/s00200-013-0207-y","volume":"24","author":"N Aydin","year":"2013","unstructured":"Aydin, N., Karadeniz, S., Yildiz, B.: Some new binary quasi-cyclic codes from codes over the ring \n                    \n                      \n                    \n                    $${\\mathbb{F}}_2+u{\\mathbb{F}}_2+v{\\mathbb{F}}_2+uv{\\mathbb{F}}_2$$\n                    \n                      \n                        \n                          \n                            F\n                            2\n                          \n                          +\n                          u\n                          \n                            F\n                            2\n                          \n                          +\n                          v\n                          \n                            F\n                            2\n                          \n                          +\n                          u\n                          v\n                          \n                            F\n                            2\n                          \n                        \n                      \n                    \n                  . Appl. Algebra Eng. Commun. Comput. 24(5), 355\u2013367 (2013). \n                    https:\/\/doi.org\/10.1007\/s00200-013-0207-y","journal-title":"Appl. Algebra Eng. Commun. Comput."},{"key":"384_CR2","doi-asserted-by":"publisher","first-page":"188","DOI":"10.1016\/j.dam.2013.11.017","volume":"167","author":"ST Dougherty","year":"2014","unstructured":"Dougherty, S.T., Karadeniz, S., Yildiz, B.: Constructing formally self-dual codes over \n                    \n                      \n                    \n                    $$R_k$$\n                    \n                      \n                        \n                          R\n                          k\n                        \n                      \n                    \n                  . Discrete Appl. Math. 167, 188\u2013196 (2014). \n                    https:\/\/doi.org\/10.1016\/j.dam.2013.11.017","journal-title":"Discrete Appl. Math."},{"issue":"1","key":"384_CR3","doi-asserted-by":"publisher","first-page":"55","DOI":"10.1007\/s10623-008-9243-1","volume":"51","author":"ST Dougherty","year":"2009","unstructured":"Dougherty, S.T., Liu, H.: Independence of vectors in codes over rings. Des. Codes Cryptogr. 51(1), 55\u201368 (2009). \n                    https:\/\/doi.org\/10.1007\/s10623-008-9243-1","journal-title":"Des. Codes Cryptogr."},{"issue":"2","key":"384_CR4","doi-asserted-by":"publisher","first-page":"379","DOI":"10.3934\/amc.2016012","volume":"10","author":"ST Dougherty","year":"2016","unstructured":"Dougherty, S.T., Salt\u00fcrk, E., Szabo, S.: Codes over local rings of order 16 and binary codes. Adv. Math. Commun. 10(2), 379\u2013391 (2016). \n                    https:\/\/doi.org\/10.3934\/amc.2016012","journal-title":"Adv. Math. Commun."},{"key":"384_CR5","doi-asserted-by":"publisher","unstructured":"Mart\u00ednez-Moro, E., Szabo, S.: On codes over local Frobenius non-chain rings of order 16. In: Dougherty, S., Facchini, A., Leroy, A., Puczy\u0142owski, E., Sol\u00e9, P. (eds.) Noncommutative rings and their applications. Contemporary Mathematics, vol. 634, pp. 227\u2013241. American Mathematical Society, Providence (2015). \n                    https:\/\/doi.org\/10.1090\/conm\/634\/12702","DOI":"10.1090\/conm\/634\/12702"},{"issue":"1","key":"384_CR6","doi-asserted-by":"publisher","first-page":"78","DOI":"10.1504\/IJICOT.2015.068698","volume":"3","author":"E Mart\u00ednez-Moro","year":"2015","unstructured":"Mart\u00ednez-Moro, E., Szabo, S., Yildiz, B.: Linear codes over \n                    \n                      \n                    \n                    $$\\frac{{\\mathbb{Z}}_4[x]}{\\langle x^2+2x\\rangle }$$\n                    \n                      \n                        \n                          \n                            \n                              Z\n                              4\n                            \n                            \n                              [\n                              x\n                              ]\n                            \n                          \n                          \n                            \u27e8\n                            \n                              x\n                              2\n                            \n                            +\n                            2\n                            x\n                            \u27e9\n                          \n                        \n                      \n                    \n                  . Int. J. Inf. Coding Theory 3(1), 78\u201396 (2015)","journal-title":"Int. J. Inf. Coding Theory"},{"key":"384_CR7","volume-title":"Finite Rings with Identity","author":"BR McDonald","year":"1974","unstructured":"McDonald, B.R.: Finite Rings with Identity. Marcel Dekker Inc, New York (1974)"},{"issue":"3","key":"384_CR8","doi-asserted-by":"publisher","first-page":"555","DOI":"10.1353\/ajm.1999.0024","volume":"121","author":"JA Wood","year":"1999","unstructured":"Wood, J.A.: Duality for modules over finite rings and applications to coding theory. Am. J. Math. 121(3), 555\u2013575 (1999)","journal-title":"Am. J. Math."},{"issue":"1","key":"384_CR9","doi-asserted-by":"publisher","first-page":"61","DOI":"10.1007\/s10623-009-9309-8","volume":"54","author":"B Yildiz","year":"2010","unstructured":"Yildiz, B., Karadeniz, S.: Linear codes over \n                    \n                      \n                    \n                    $${\\mathbb{F}}_2+u{\\mathbb{F}}_2+v{\\mathbb{F}}_2+uv{\\mathbb{F}}_2$$\n                    \n                      \n                        \n                          \n                            F\n                            2\n                          \n                          +\n                          u\n                          \n                            F\n                            2\n                          \n                          +\n                          v\n                          \n                            F\n                            2\n                          \n                          +\n                          u\n                          v\n                          \n                            F\n                            2\n                          \n                        \n                      \n                    \n                  . Des. Codes Cryptogr. 54(1), 61\u201381 (2010). \n                    https:\/\/doi.org\/10.1007\/s10623-009-9309-8","journal-title":"Des. Codes Cryptogr."},{"issue":"10","key":"384_CR10","doi-asserted-by":"publisher","first-page":"1888","DOI":"10.1016\/j.jfranklin.2010.10.007","volume":"347","author":"B Yildiz","year":"2010","unstructured":"Yildiz, B., Karadeniz, S.: Self-dual codes over \n                    \n                      \n                    \n                    $${\\mathbb{F}}_2+u{\\mathbb{F}}_2+v{\\mathbb{F}}_2+uv{\\mathbb{F}}_2$$\n                    \n                      \n                        \n                          \n                            F\n                            2\n                          \n                          +\n                          u\n                          \n                            F\n                            2\n                          \n                          +\n                          v\n                          \n                            F\n                            2\n                          \n                          +\n                          u\n                          v\n                          \n                            F\n                            2\n                          \n                        \n                      \n                    \n                  . J. Frankl. Inst. 347(10), 1888\u20131894 (2010). \n                    https:\/\/doi.org\/10.1016\/j.jfranklin.2010.10.007","journal-title":"J. Frankl. Inst."},{"key":"384_CR11","doi-asserted-by":"publisher","first-page":"24","DOI":"10.1016\/j.ffa.2013.12.007","volume":"27","author":"B Yildiz","year":"2014","unstructured":"Yildiz, B., Karadeniz, S.: Linear codes over \n                    \n                      \n                    \n                    $${\\mathbb{Z}}_4+u{\\mathbb{Z}}_4$$\n                    \n                      \n                        \n                          \n                            Z\n                            4\n                          \n                          +\n                          u\n                          \n                            Z\n                            4\n                          \n                        \n                      \n                    \n                  : MacWilliams identities, projections, and formally self-dual codes. Finite Fields Appl. 27, 24\u201340 (2014). \n                    https:\/\/doi.org\/10.1016\/j.ffa.2013.12.007","journal-title":"Finite Fields Appl."}],"container-title":["Applicable Algebra in Engineering, Communication and Computing"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00200-019-00384-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s00200-019-00384-0\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00200-019-00384-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,3,6]],"date-time":"2020-03-06T00:09:19Z","timestamp":1583453359000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s00200-019-00384-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,3,7]]},"references-count":11,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2019,6]]}},"alternative-id":["384"],"URL":"https:\/\/doi.org\/10.1007\/s00200-019-00384-0","relation":{},"ISSN":["0938-1279","1432-0622"],"issn-type":[{"value":"0938-1279","type":"print"},{"value":"1432-0622","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,3,7]]},"assertion":[{"value":"3 March 2015","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"20 June 2016","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"7 March 2019","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}