{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,28]],"date-time":"2025-09-28T12:44:17Z","timestamp":1759063457238},"publisher-location":"Cham","reference-count":16,"publisher":"Springer International Publishing","isbn-type":[{"type":"print","value":"9783319499826"},{"type":"electronic","value":"9783319499840"}],"license":[{"start":{"date-parts":[[2017,1,1]],"date-time":"2017-01-01T00:00:00Z","timestamp":1483228800000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2017]]},"DOI":"10.1007\/978-3-319-49984-0_5","type":"book-chapter","created":{"date-parts":[[2017,3,3]],"date-time":"2017-03-03T05:52:11Z","timestamp":1488520331000},"page":"79-91","source":"Crossref","is-referenced-by-count":3,"title":["The Dual of Convolutional Codes Over $${\\mathbb {Z}}_{p^r}$$"],"prefix":"10.1007","author":[{"given":"Mohammed","family":"El Oued","sequence":"first","affiliation":[]},{"given":"Diego","family":"Napp","sequence":"additional","affiliation":[]},{"given":"Raquel","family":"Pinto","sequence":"additional","affiliation":[]},{"given":"Marisa","family":"Toste","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2017,3,4]]},"reference":[{"key":"5_CR1","doi-asserted-by":"crossref","unstructured":"Calderbank, A.R., Sloane, N.J.A.: Modular and $$p$$ -adic cyclic codes. Des. Codes Cryptogr. 6(1), 21\u201335 (1995)","DOI":"10.1007\/BF01390768"},{"key":"5_CR2","doi-asserted-by":"crossref","first-page":"30","DOI":"10.1016\/0097-3165(93)90070-O","volume":"62","author":"JH Conway","year":"1993","unstructured":"Conway, J.H., Sloane, N.J.A.: Self-dual codes over the integers modulo 4. J. Comb. Theory A 62, 30\u201345 (1993)","journal-title":"J. Comb. Theory A"},{"issue":"5","key":"5_CR3","doi-asserted-by":"crossref","first-page":"720","DOI":"10.1109\/TIT.1970.1054541","volume":"IT\u201316","author":"GD Forney Jr","year":"1970","unstructured":"Forney Jr., G.D.: Convolutional codes I: algebraic structure. IEEE Trans. Inf. Theory IT\u201316(5), 720\u2013738 (1970)","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"5","key":"5_CR4","doi-asserted-by":"crossref","first-page":"512","DOI":"10.1109\/TIT.1973.1055030","volume":"IT\u201319","author":"GD Forney Jr","year":"1973","unstructured":"Forney Jr., G.D.: Structural analysis of convolutional codes via dual codes. Trans. Inf. Theory IT\u201319(5), 512\u2013518 (1973)","journal-title":"Trans. Inf. Theory"},{"key":"5_CR5","volume-title":"Basic Algebra II","author":"N Jacobson","year":"1989","unstructured":"Jacobson, N.: Basic Algebra II. W. H. Freeman, San Francisco (1989)"},{"issue":"11","key":"5_CR6","first-page":"4890","volume":"55","author":"M Kuijper","year":"2009","unstructured":"Kuijper, M., Pinto, R.: On minimality of convolutional ring encoders. IEEE Trans. Autom. Control 55(11), 4890\u20134897 (2009)","journal-title":"IEEE Trans. Autom. Control"},{"issue":"2\u20133","key":"5_CR7","doi-asserted-by":"crossref","first-page":"776","DOI":"10.1016\/j.laa.2007.04.015","volume":"425","author":"M Kuijper","year":"2007","unstructured":"Kuijper, M., Pinto, R., Polderman, J.W.: The predictable degree property and row reducedness for systems over a finite ring. Linear Algebr. Appl. 425(2\u20133), 776\u2013796 (2007)","journal-title":"Linear Algebr. Appl."},{"key":"5_CR8","unstructured":"Massey, J. L., Mittelholzer, T.: Convolutional codes over rings. In: Proceedings of the 4th Joint Swedish-Soviet International Workshop Information Theory, pp. 14\u201318 (1989)"},{"key":"5_CR9","first-page":"1065","volume-title":"Handbook of Coding Theory","author":"RJ McEliece","year":"1998","unstructured":"McEliece, R.J.: The algebraic theory of convolutional codes. Handbook of Coding Theory, vol. 1, pp. 1065\u20131138. Elsevier Science Publishers, Amsterdam (1998)"},{"key":"5_CR10","unstructured":"Napp, D., Pinto, R., Toste, M.: On MDS convolutional codes over $${\\mathbb{Z}}_{{p}^{r}}$$ . http:\/\/arxiv.org\/abs\/1601.04507 . (Submitted on 18 Jan 2016)"},{"issue":"6","key":"5_CR11","doi-asserted-by":"crossref","first-page":"489","DOI":"10.1007\/PL00012382","volume":"10","author":"G Norton","year":"2000","unstructured":"Norton, G., Salagean, A.: On the structure of linear and cyclic codes over a finite chain ring. Appl. Algebr. Eng. Commun. Comput. 10(6), 489\u2013506 (2000)","journal-title":"Appl. Algebr. Eng. Commun. Comput."},{"issue":"3","key":"5_CR12","doi-asserted-by":"crossref","first-page":"1060","DOI":"10.1109\/18.841186","volume":"46","author":"GH Norton","year":"2001","unstructured":"Norton, G.H., Salagean, A.: On the hamming distance of linear codes over a finite chain ring. IEEE Trans. Inf. Theory 46(3), 1060\u20131067 (2001)","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"2","key":"5_CR13","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1504\/IJICOT.2015.072612","volume":"3","author":"M El Oued","year":"2015","unstructured":"Oued, M.El: On MDR codes over a finite ring. IJICoT 3(2), 107\u2013119 (2015)","journal-title":"IJICoT"},{"issue":"11","key":"5_CR14","doi-asserted-by":"crossref","first-page":"7305","DOI":"10.1109\/TIT.2013.2277721","volume":"59","author":"M El Oued","year":"2013","unstructured":"Oued, M.El, Sol\u00e9, P.: MDS convolutional codes over a finite ring. IEEE Trans. Inf. Theory 59(11), 7305\u20137313 (2013)","journal-title":"IEEE Trans. Inf. Theory"},{"key":"5_CR15","doi-asserted-by":"crossref","first-page":"1839","DOI":"10.1109\/18.556679","volume":"42","author":"VV Vazirani","year":"1996","unstructured":"Vazirani, V.V., Saran, H., Rajan, B.S.: An efficient algorithm for constructing minimal trellises for codes over finite abelian groups. IEEE Trans. Inf. Theory 42, 1839\u20131854 (1996)","journal-title":"IEEE Trans. Inf. Theory"},{"key":"5_CR16","doi-asserted-by":"crossref","unstructured":"Wood, J.: Duality for modules over finite rings and applications to coding theory. Am. J. Math. 121(3), 555\u2013575 (1999)","DOI":"10.1353\/ajm.1999.0024"}],"container-title":["Springer Proceedings in Mathematics &amp; Statistics","Applied and Computational Matrix Analysis"],"original-title":[],"link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/978-3-319-49984-0_5","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2017,6,25]],"date-time":"2017-06-25T11:49:03Z","timestamp":1498391343000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/978-3-319-49984-0_5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017]]},"ISBN":["9783319499826","9783319499840"],"references-count":16,"URL":"https:\/\/doi.org\/10.1007\/978-3-319-49984-0_5","relation":{},"ISSN":["2194-1009","2194-1017"],"issn-type":[{"type":"print","value":"2194-1009"},{"type":"electronic","value":"2194-1017"}],"subject":[],"published":{"date-parts":[[2017]]}}}