{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,11]],"date-time":"2026-03-11T18:20:50Z","timestamp":1773253250073,"version":"3.50.1"},"reference-count":26,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2009,11,6]],"date-time":"2009-11-06T00:00:00Z","timestamp":1257465600000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Des. Codes Cryptogr."],"published-print":{"date-parts":[[2010,7]]},"DOI":"10.1007\/s10623-009-9340-9","type":"journal-article","created":{"date-parts":[[2009,11,5]],"date-time":"2009-11-05T05:40:52Z","timestamp":1257399652000},"page":"43-59","source":"Crossref","is-referenced-by-count":38,"title":["$${\\mathbb{Z}_2\\mathbb{Z}_4}$$ -linear codes: rank and kernel"],"prefix":"10.1007","volume":"56","author":[{"given":"Cristina","family":"Fern\u00e1ndez-C\u00f3rdoba","sequence":"first","affiliation":[]},{"given":"Jaume","family":"Pujol","sequence":"additional","affiliation":[]},{"given":"Merc\u00e8","family":"Villanueva","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2009,11,6]]},"reference":[{"key":"9340_CR1","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1007\/BF02579339","volume":"3","author":"H. Bauer","year":"1983","unstructured":"Bauer H., Ganter B., Hergert F.: Algebraic techniques for nonlinear codes. Combinatorica 3, 21\u201333 (1983)","journal-title":"Combinatorica"},{"key":"9340_CR2","volume-title":"Introduction to coding theory","author":"J. Bierbrauer","year":"2005","unstructured":"Bierbrauer J.: Introduction to coding theory. Chapman & Hall\/CRC, Boca Raton (2005)"},{"issue":"7","key":"9340_CR3","doi-asserted-by":"crossref","first-page":"2686","DOI":"10.1109\/TIT.2005.850148","volume":"51","author":"J. Borges","year":"2005","unstructured":"Borges J., Fern\u00e1ndez C., Phelps K.T.: Quaternary Reed\u2013Muller codes. IEEE Trans. Inform. Theory. 51(7), 2686\u20132691 (2005)","journal-title":"IEEE Trans. Inform. Theory."},{"issue":"1","key":"9340_CR4","doi-asserted-by":"crossref","first-page":"380","DOI":"10.1109\/TIT.2007.911211","volume":"54","author":"J. Borges","year":"2008","unstructured":"Borges J., Fern\u00e1ndez-C\u00f3rdoba C., Phelps K.T.: ZRM codes. IEEE Trans. Inform. Theory. 54(1), 380\u2013386 (2008)","journal-title":"IEEE Trans. Inform. Theory."},{"key":"9340_CR5","unstructured":"Borges J., Fern\u00e1ndez C., Pujol J., Rif\u00e0 J., Villanueva M.: $${{\\mathbb{Z}_2\\mathbb{Z}_4}}$$ -linear codes: generator matrices and duality, to appear in designs, codes and cryptography. arXiv:0710.1149 (2009)."},{"key":"9340_CR6","unstructured":"Borges J., Fern\u00e1ndez C., Pujol J., Rif\u00e0 J., Villanueva M.: $${{\\mathbb{Z}_2\\mathbb{Z}_4}}$$ -additive codes. A Magma package. Universitat Aut\u00f2noma de Barcelona. http:\/\/www.ccg.uab.cat (2007)."},{"issue":"8","key":"9340_CR7","doi-asserted-by":"crossref","first-page":"2028","DOI":"10.1109\/TIT.2003.814490","volume":"49","author":"J. Borges","year":"2003","unstructured":"Borges J., Phelps K.T., Rif\u00e0 J.: The rank and kernel of extended 1-perfect $${\\mathbb{Z}_4}$$ -linear and additive non- $${\\mathbb{Z}_4}$$ \u2013linear codes. IEEE Trans. Inform. Theory. 49(8), 2028\u20132034 (2003)","journal-title":"IEEE Trans. Inform. Theory."},{"issue":"11","key":"9340_CR8","doi-asserted-by":"crossref","first-page":"2834","DOI":"10.1109\/TIT.2003.819329","volume":"49","author":"J. Borges","year":"2003","unstructured":"Borges J., Phelps K.T., Rif\u00e0 J., Zinoviev V.A.: On $${\\mathbb{Z}_4}$$ -linear preparata-like and kerdock-like codes. IEEE Trans. Inform. Theory. 49(11), 2834\u20132843 (2003)","journal-title":"IEEE Trans. Inform. Theory."},{"issue":"5","key":"9340_CR9","doi-asserted-by":"crossref","first-page":"1688","DOI":"10.1109\/18.771247","volume":"45","author":"J. Borges","year":"1999","unstructured":"Borges J., Rif\u00e0 J.: A characterization of 1-perfect additive codes. IEEE Trans. Inform. Theory. 45(5), 1688\u20131697 (1999)","journal-title":"IEEE Trans. Inform. Theory."},{"key":"9340_CR10","unstructured":"Cannon J.J., Bosma W.: (eds.) Handbook of Magma functions, edn. 2.13. 4350\u00a0pp. University of Kashan, Kashan, Iran (2006)."},{"issue":"2","key":"9340_CR11","first-page":"1","volume":"10","author":"P. Delsarte","year":"1973","unstructured":"Delsarte P.: An algebraic approach to the association schemes of coding theory. Philips Res. Rep. Suppl. 10(2), 1\u201397 (1973)","journal-title":"Philips Res. Rep. Suppl."},{"issue":"6","key":"9340_CR12","doi-asserted-by":"crossref","first-page":"2477","DOI":"10.1109\/18.720545","volume":"44","author":"P. Delsarte","year":"1998","unstructured":"Delsarte P., Levenshtein V.: Asociation schemes and coding theory. IEEE Trans. Inform. Theory. 44(6), 2477\u20132504 (1998)","journal-title":"IEEE Trans. Inform. Theory."},{"key":"9340_CR13","unstructured":"Fern\u00e1ndez C., Villanueva M.: $${{\\mathbb{Z}_2\\mathbb{Z}_4}}$$ -linear codes with all possible ranks and kernel dimensions, in XI international symposium on problems of redundancy in information and control systems. St. Petersburg, Russia, pp. 62\u201365, July (2007)."},{"key":"9340_CR14","doi-asserted-by":"crossref","unstructured":"Fern\u00e1ndez-C\u00f3rdoba C., Pujol J., Villanueva M.: On rank and kernel of $${\\mathbb{Z}_4}$$ -linear codes. Lecture Notes in Computer Science no. 5228, pp. 46\u201355 (2008).","DOI":"10.1007\/978-3-540-87448-5_6"},{"key":"9340_CR15","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1109\/18.312154","volume":"40","author":"A.R. Hammons","year":"1994","unstructured":"Hammons A.R., Kumar P.V., Calderbank A.R., Sloane N.J.A., Sol\u00e9 P.: The $${\\mathbb{Z}_4}$$ -linearity of kerdock, preparata, goethals and related codes. IEEE Trans. Inform. Theory. 40, 301\u2013319 (1994)","journal-title":"IEEE Trans. Inform. Theory."},{"key":"9340_CR16","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511807077","volume-title":"Fundamentals of error-correcting codes","author":"W.C. Huffman","year":"2003","unstructured":"Huffman W.C., Pless V.: Fundamentals of error-correcting codes. Cambridge University Press, Cambridge (2003)"},{"key":"9340_CR17","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1016\/S1571-0653(04)00161-1","volume":"6","author":"D.S. Krotov","year":"2001","unstructured":"Krotov D.S.: $${\\mathbb{Z}_4}$$ -linear Hadamard and extended perfect codes. Electron. Note Discr. Math. 6, 107\u2013112 (2001)","journal-title":"Electron. Note Discr. Math."},{"key":"9340_CR18","doi-asserted-by":"crossref","first-page":"57","DOI":"10.4153\/CMB-1975-011-2","volume":"18","author":"B. Lindstr\u00f6m","year":"1975","unstructured":"Lindstr\u00f6m B.: Group partitions and mixed perfect codes. Canad. Math. Bull. 18, 57\u201360 (1975)","journal-title":"Canad. Math. Bull."},{"key":"9340_CR19","volume-title":"The theory of error correcting codes","author":"F.J. MacWillams","year":"1977","unstructured":"MacWillams F.J., Sloane N.J.A.: The theory of error correcting codes. North Holland, Amsterdam (1977)"},{"key":"9340_CR20","doi-asserted-by":"crossref","unstructured":"Pernas J., Pujol J., Villanueva M.: Kernel dimension for some families of quaternary Reed\u2013Muller codes. Lecture Notes in Computer Science n. 5393. pp. 128\u2013141 (2008).","DOI":"10.1007\/978-3-540-89994-5_11"},{"issue":"3","key":"9340_CR21","doi-asserted-by":"crossref","first-page":"247","DOI":"10.1007\/BF01388478","volume":"6","author":"K.T. Phelps","year":"1995","unstructured":"Phelps K.T., LeVan M.: Kernels of nonlinear Hamming codes. Design code cryptogr. 6(3), 247\u2013257 (1995)","journal-title":"Design code cryptogr."},{"issue":"9","key":"9340_CR22","doi-asserted-by":"crossref","first-page":"2587","DOI":"10.1109\/TIT.2002.801474","volume":"48","author":"K.T. Phelps","year":"2002","unstructured":"Phelps K.T., Rif\u00e0 J.: On binary 1-perfect additive codes: some structural properties. IEEE Trans. Inform. Theory. 48(9), 2587\u20132592 (2002)","journal-title":"IEEE Trans. Inform. Theory."},{"issue":"1","key":"9340_CR23","doi-asserted-by":"crossref","first-page":"316","DOI":"10.1109\/TIT.2005.860464","volume":"52","author":"K.T. Phelps","year":"2005","unstructured":"Phelps K.T., Rif\u00e0 J., Villanueva M.: On the additive $${\\mathbb{Z}_4}$$ -linear and non- $${\\mathbb{Z}_4}$$ -linear Hadamard codes. Rank and Kernel. IEEE Trans. Inform. Theory. 52(1), 316\u2013319 (2005)","journal-title":"Rank and Kernel. IEEE Trans. Inform. Theory."},{"key":"9340_CR24","doi-asserted-by":"crossref","first-page":"590","DOI":"10.1109\/18.556115","volume":"43","author":"J. Pujol","year":"1997","unstructured":"Pujol J., Rif\u00e0 J.: Translation invariant propelinear codes. IEEE Trans. Inform. Theory. 43, 590\u2013598 (1997)","journal-title":"IEEE Trans. Inform. Theory."},{"issue":"1","key":"9340_CR25","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1109\/TIT.2008.2008143","volume":"55","author":"J. Pujol","year":"2009","unstructured":"Pujol J., Rif\u00e0 J., Solov\u2019eva F.I.: Construction of $${\\mathbb{Z}_4}$$ -linear Reed\u2013Muller codes. IEEE Trans. Inform. Theory. 55(1), 99\u2013104 (2009)","journal-title":"IEEE Trans. Inform. Theory."},{"key":"9340_CR26","doi-asserted-by":"crossref","DOI":"10.1142\/3603","volume-title":"Quaternary codes","author":"Z.-X. Wan","year":"1997","unstructured":"Wan Z.-X.: Quaternary codes. World Scientific, Singapore (1997)"}],"container-title":["Designs, Codes and Cryptography"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10623-009-9340-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10623-009-9340-9\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10623-009-9340-9","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,30]],"date-time":"2019-05-30T19:58:27Z","timestamp":1559246307000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10623-009-9340-9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,11,6]]},"references-count":26,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2010,7]]}},"alternative-id":["9340"],"URL":"https:\/\/doi.org\/10.1007\/s10623-009-9340-9","relation":{},"ISSN":["0925-1022","1573-7586"],"issn-type":[{"value":"0925-1022","type":"print"},{"value":"1573-7586","type":"electronic"}],"subject":[],"published":{"date-parts":[[2009,11,6]]}}}