{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,3,26]],"date-time":"2025-03-26T19:32:46Z","timestamp":1743017566865,"version":"3.40.3"},"publisher-location":"Cham","reference-count":20,"publisher":"Springer International Publishing","isbn-type":[{"type":"print","value":"9783319172958"},{"type":"electronic","value":"9783319172965"}],"license":[{"start":{"date-parts":[[2015,1,1]],"date-time":"2015-01-01T00:00:00Z","timestamp":1420070400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2015,1,1]],"date-time":"2015-01-01T00:00:00Z","timestamp":1420070400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2015]]},"DOI":"10.1007\/978-3-319-17296-5_18","type":"book-chapter","created":{"date-parts":[[2015,7,24]],"date-time":"2015-07-24T14:30:01Z","timestamp":1437748201000},"page":"177-183","source":"Crossref","is-referenced-by-count":4,"title":["The Extension Theorem with Respect to Symmetrized Weight Compositions"],"prefix":"10.1007","author":[{"given":"Noha","family":"ElGarem","sequence":"first","affiliation":[]},{"given":"Nefertiti","family":"Megahed","sequence":"additional","affiliation":[]},{"given":"Jay A.","family":"Wood","sequence":"additional","affiliation":[]}],"member":"297","reference":[{"key":"18_CR1","unstructured":"Barra, A., Gluesing-Luerssen, H.: MacWilliams extension theorems and the local-global property for codes over rings (2013). arXiv:1307.7159"},{"issue":"3","key":"18_CR2","first-page":"208","volume":"33","author":"I. Constantinescu","year":"1997","unstructured":"Constantinescu, I., Heise, W.: A metric for codes over residue class rings of integers. Probl. Inf. Trans. 33(3), 208\u2013213 (1997)","journal-title":"Probl. Inf. Trans."},{"key":"18_CR3","unstructured":"Constantinescu, I., Heise, W., Honold, T.: Monomial extensions of isometries between codes over \n$$\\mathbb{Z}_{m}$$\n. In: Proceedings of the Fifth International Workshop on Algebraic and Combinatorial Coding Theory, Sozopol, pp.\u00a098\u2013104. Unicorn, Shumen (1996)"},{"issue":"4","key":"18_CR4","doi-asserted-by":"publisher","first-page":"615","DOI":"10.1016\/j.ffa.2004.01.001","volume":"10","author":"H.Q. Dinh","year":"2004","unstructured":"Dinh, H.Q., L\u00f3pez-Permouth, S.R.: On the equivalence of codes over rings and modules. Finite Fields Appl. 10(4), 615\u2013625 (2004). doi:10.1016\/j.ffa.2004.01.001","journal-title":"Finite Fields Appl."},{"issue":"3","key":"18_CR5","doi-asserted-by":"publisher","first-page":"363","DOI":"10.1016\/0097-3165(80)90032-1","volume":"29","author":"D. Goldberg","year":"1980","unstructured":"Goldberg, D.: A generalized weight for linear codes and a Witt-MacWilliams theorem. J. Comb. Theory Ser. A 29(3), 363\u2013367 (1980)","journal-title":"J. Comb. Theory Ser. A"},{"key":"18_CR6","unstructured":"Greferath, M., Honold, T.: On weights allowing for MacWilliams equivalence theorem. In: Proceedings of the Fourth International Workshop on Optimal Codes and Related Topics, Pamporovo, pp.\u00a0182\u2013192 (2005)"},{"key":"18_CR7","unstructured":"Greferath, M., Honold, T.: Monomial extensions of isometries of linear codes ii: invariant weight functions on \n$$\\mathbb{Z}_{m}$$\n. In: Proceedings of the Tenth International Workshop on Algebraic and Combinatorial Coding Theory, Zvenigorod, pp.\u00a0106\u2013111 (2006)"},{"key":"18_CR8","doi-asserted-by":"publisher","first-page":"177","DOI":"10.1016\/j.jcta.2014.03.005","volume":"125","author":"M. Greferath","year":"2014","unstructured":"Greferath, M., Honold, T., McFadden, C., Wood, J.A., Zumbr\u00e4gel, J.: MacWilliams\u2019 extension theorem for bi-invariant weights over finite principal ideal rings. J. Comb. Theory Ser. A 125, 177\u2013193 (2014)","journal-title":"J. Comb. Theory Ser. A"},{"issue":"1\u20133","key":"18_CR9","doi-asserted-by":"publisher","first-page":"145","DOI":"10.1007\/s10623-012-9671-9","volume":"66","author":"M. Greferath","year":"2013","unstructured":"Greferath, M., McFadden, C., Zumbr\u00e4gel, J.: Characteristics of invariant weights related to code equivalence over rings. Des. Codes Cryptogr. 66(1\u20133), 145\u2013156 (2013)","journal-title":"Des. Codes Cryptogr."},{"issue":"3","key":"18_CR10","doi-asserted-by":"publisher","first-page":"247","DOI":"10.1142\/S0219498804000873","volume":"3","author":"M. Greferath","year":"2004","unstructured":"Greferath, M., Nechaev, R., Wisbauer, R.: Finite quasi-Frobenius modules and linear codes. J. Algebra Appl. 3(3), 247\u2013272 (2004)","journal-title":"J. Algebra Appl."},{"issue":"1","key":"18_CR11","doi-asserted-by":"publisher","first-page":"17","DOI":"10.1006\/jcta.1999.3033","volume":"92","author":"M. Greferath","year":"2000","unstructured":"Greferath, M., Schmidt, S.E.: Finite ring combinatorics and MacWilliams equivalence theorem. J. Comb. Theory Ser. A 92(1), 17\u201328 (2000). doi:10.1006\/jcta.1999.3033","journal-title":"J. Comb. Theory Ser. A"},{"issue":"2","key":"18_CR12","doi-asserted-by":"publisher","first-page":"301","DOI":"10.1109\/18.312154","volume":"40","author":"A. Hammons","year":"1994","unstructured":"Hammons, A., Kumar, P., Calderbank, A., Sloane, N., Sol\u00e9, P.: The \n$$\\mathbb{Z}_{4}$$\n linearity of Kerdock, Preparata, Goethals, and related codes. IEEE Trans. Inf. Theory 40(2), 301\u2013319 (1994). doi:10.1109\/18.312154","journal-title":"IEEE Trans. Inf. Theory"},{"key":"18_CR13","unstructured":"MacWilliams, F.J.: Combinatorial properties of elementary abelian groups. Ph.D. thesis, Harvard University, Cambridge (1962)"},{"key":"18_CR14","doi-asserted-by":"crossref","unstructured":"Ward, H.N., Wood, J.A.: Characters and the equivalence of codes. J. Comb. Theory Ser. A 73(2), 348\u2013352 (1996). doi:10.1016\/S0097-3165(96)80011-2","DOI":"10.1016\/S0097-3165(96)80011-2"},{"key":"18_CR15","doi-asserted-by":"publisher","first-page":"329","DOI":"10.1007\/3-540-63163-1_26","volume-title":"Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. Lecture Notes in Computer Science","author":"J.A. Wood","year":"1997","unstructured":"Wood, J.A.: Extension theorems for linear codes over finite rings. In: Mora, T., Mattson, H. (eds.) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. Lecture Notes in Computer Science, vol.\u00a01255, pp.\u00a0329\u2013340. Springer, Berlin\/Heidelberg (1997). doi:10.1007\/3-540-63163-1_26"},{"key":"18_CR16","unstructured":"Wood, J.A.: Weight functions and the extension theorem for linear codes over finite rings. In: Proceedings of the Fourth International Conference on Finite Fields: Theory, Applications and Algorithms, University of Waterloo, Waterloo (1997)"},{"key":"18_CR17","doi-asserted-by":"publisher","first-page":"555","DOI":"10.1353\/ajm.1999.0024","volume":"121","author":"J.A. Wood","year":"1999","unstructured":"Wood, J.A.: Duality for modules over finite rings and applications to coding theory. Am. J. Math. 121, 555\u2013575 (1999)","journal-title":"Am. J. Math."},{"key":"18_CR18","doi-asserted-by":"publisher","first-page":"699","DOI":"10.1090\/S0002-9939-07-09164-2","volume":"136","author":"J.A. Wood","year":"2008","unstructured":"Wood, J.A.: Code equivalence characterizes finite Frobenius rings. Proc. Am. Math. Soc. 136, 699\u2013706 (2008). doi:10.1090\/S0002-9939-07-09164-2","journal-title":"Proc. Am. Math. Soc."},{"key":"18_CR19","doi-asserted-by":"crossref","unstructured":"Wood, J.A.: Foundations of linear codes defined over finite modules: the extension theorem and MacWilliams identities. In: Sol\u00e9, P. (ed.) CIMPA Summer School, Ankara, 18\u201329 Aug 2008. Series on Coding Theory and Cryptology, vol.\u00a06, pp.\u00a0124\u2013190. World Scientific (2009)","DOI":"10.1142\/9789812837691_0004"},{"key":"18_CR20","first-page":"235","volume-title":"44th Symposium on Ring Theory and Representation Theory","author":"J.A. Wood","year":"2012","unstructured":"Wood, J.A.: Applications of finite Frobenius rings to the foundations of algebraic coding theory. In: Iyama, O. (ed.) 44th Symposium on Ring Theory and Representation Theory, Okayama University, Nagoya, pp.\u00a0235\u2013245 (2012)"}],"container-title":["CIM Series in Mathematical Sciences","Coding Theory and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/978-3-319-17296-5_18","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,1,28]],"date-time":"2023-01-28T12:23:42Z","timestamp":1674908622000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/978-3-319-17296-5_18"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015]]},"ISBN":["9783319172958","9783319172965"],"references-count":20,"URL":"https:\/\/doi.org\/10.1007\/978-3-319-17296-5_18","relation":{},"ISSN":["2364-950X","2364-9518"],"issn-type":[{"type":"print","value":"2364-950X"},{"type":"electronic","value":"2364-9518"}],"subject":[],"published":{"date-parts":[[2015]]}}}