{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,20]],"date-time":"2026-03-20T16:04:45Z","timestamp":1774022685810,"version":"3.50.1"},"reference-count":32,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2016,6,11]],"date-time":"2016-06-11T00:00:00Z","timestamp":1465603200000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Cryptogr. Commun."],"published-print":{"date-parts":[[2017,1]]},"DOI":"10.1007\/s12095-016-0194-5","type":"journal-article","created":{"date-parts":[[2016,6,11]],"date-time":"2016-06-11T08:39:31Z","timestamp":1465634371000},"page":"41-53","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":18,"title":["Optimal codebooks achieving the Levenshtein bound from generalized bent functions over \u2124 4 $\\mathbb {Z}_{4}$"],"prefix":"10.1007","volume":"9","author":[{"given":"Ziling","family":"Heng","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Qin","family":"Yue","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2016,6,11]]},"reference":[{"issue":"2","key":"194_CR1","doi-asserted-by":"crossref","first-page":"368","DOI":"10.2307\/1970804","volume":"95","author":"EH Brown","year":"1972","unstructured":"Brown, E.H.: Generalizations of the Kervaire invariant. Annals. Math. 95(2), 368\u2013383 (1972)","journal-title":"Annals. Math."},{"key":"194_CR2","doi-asserted-by":"crossref","DOI":"10.1007\/978-0-8176-8224-8","volume-title":"An Introduction to Frames and Risez Bases","author":"O Christensen","year":"2003","unstructured":"Christensen, O.: An Introduction to Frames and Risez Bases. Birkh\u00e4user, MA, USA (2003)"},{"key":"194_CR3","doi-asserted-by":"crossref","unstructured":"Calderbank A.R., Cameron, P.J., Kantor, W.M., Seidel, J.J.: \u2124 4 $\\mathbb {Z}_{4}$ -kerdock codes, orthogonal spreads, and extremal Euclidean line-sets. Proc. London Math. Soc. 75(3), 436\u2013480 (1997)","DOI":"10.1112\/S0024611597000403"},{"issue":"2","key":"194_CR4","doi-asserted-by":"crossref","first-page":"139","DOI":"10.1080\/10586458.1996.10504585","volume":"5","author":"JH Conway","year":"1996","unstructured":"Conway, J.H., Harding, R.H., Sloane, N.J.A.: Packing lines, planes, etc.: Packings in grassmannian spaces. Exp. Math. 5(2), 139\u2013159 (1996)","journal-title":"Exp. Math."},{"key":"194_CR5","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1007\/s10623-015-0145-8","volume":"78","author":"C Carlet","year":"2016","unstructured":"Carlet, C., Mesnager, S.: Four decades of research on bent functions. Des. Codes Crypt. 78, 5\u201350 (2016)","journal-title":"Des. Codes Crypt."},{"issue":"9","key":"194_CR6","doi-asserted-by":"crossref","first-page":"4229","DOI":"10.1109\/TIT.2006.880058","volume":"52","author":"C Ding","year":"2006","unstructured":"Ding, C.: Complex codebooks from combinatorial designs. IEEE Trans. Inf. Theory 52(9), 4229\u20134235 (2006)","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"11","key":"194_CR7","first-page":"4245","volume":"53","author":"C Ding","year":"2007","unstructured":"Ding, C., Feng, T.: Codebooks from almost difference sets. Des. codes Crypt. 53(11), 4245\u20134250 (2007)","journal-title":"Des. codes Crypt."},{"issue":"1","key":"194_CR8","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1007\/s10623-007-9140-z","volume":"46","author":"C Ding","year":"2008","unstructured":"Ding, C., Feng, T.: A generic construction of complex codebooks meeting the Welch bound. Des. Codes Crypt. 46(1), 113\u2013126 (2008)","journal-title":"Des. Codes Crypt."},{"issue":"5","key":"194_CR9","doi-asserted-by":"crossref","first-page":"936","DOI":"10.1109\/TCOMM.2007.894113","volume":"55","author":"C Ding","year":"2007","unstructured":"Ding, C., Yin, J.: Singal sets from functions with optimal nonlinearity. IEEE Trans. Commun. 55(5), 936\u2013940 (2007)","journal-title":"IEEE Trans. Commun."},{"key":"194_CR10","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1016\/j.ffa.2015.06.006","volume":"36","author":"Z Heng","year":"2015","unstructured":"Heng, Z., Yue, Q.: Generalized Gray map and a class of p-ary nonlinear codes. Finite Fields Appl. 36, 81\u201397 (2015)","journal-title":"Finite Fields Appl."},{"issue":"2","key":"194_CR11","doi-asserted-by":"crossref","first-page":"233","DOI":"10.1007\/s10623-012-9648-8","volume":"69","author":"L Hu","year":"2013","unstructured":"Hu, L., Yue, Q.: Gauss periods and codebooks from generalized cyclotomic sets of order four. Des. Codes Crypt. 69(2), 233\u2013246 (2013)","journal-title":"Des. Codes Crypt."},{"issue":"3","key":"194_CR12","first-page":"257","volume":"9","author":"T Helleseth","year":"1996","unstructured":"Helleseth, T., Kumar, P.V.: Codes with the same weight distribution as the Goethals codes and the Delsarte-Goethals codes. Des. Codes Crypt. 9(3), 257\u2013266 (1996)","journal-title":"Des. Codes Crypt."},{"key":"194_CR13","doi-asserted-by":"crossref","unstructured":"Hammons, Jr. A.R., Kumar, P.V., Calderbank, A.R., Sloane, N.J.A., Sol\u00e9, P.: The \u2124 4 $\\mathbb {Z}_{4}$ -linearity of Kerdock, Preparata, Goethals, and related codes. IEEE Trans. Inf. Theory 40(2), 301\u2013319 (1994)","DOI":"10.1109\/18.312154"},{"issue":"1","key":"194_CR14","doi-asserted-by":"crossref","first-page":"458","DOI":"10.1109\/TIT.2008.2008122","volume":"55","author":"W Jiang","year":"2009","unstructured":"Jiang, W., Hu, L., Tang, X., Zeng, X.: New optimal quadriphase sequences with larger linear span. IEEE Trans. Inf. Theory 55(1), 458\u2013470 (2009)","journal-title":"IEEE Trans. Inf. Theory"},{"key":"194_CR15","first-page":"1","volume":"14","author":"GA Kabatyanskii","year":"1978","unstructured":"Kabatyanskii, G.A., Levenshtein, V.I.: Bounds for packing on a sphere and in space. Probl. Inf. Transmission 14, 1\u201317 (1978)","journal-title":"Probl. Inf. Transmission"},{"issue":"111","key":"194_CR16","first-page":"3059","volume":"149","author":"SH Kim","year":"2003","unstructured":"Kim, S.H., No, J.S.: New families of binary sequences with low correlation. IEEE Trans. Inf. Theory 149(111), 3059\u20133065 (2003)","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"1","key":"194_CR17","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1007\/s10623-013-9891-7","volume":"75","author":"C Li","year":"2015","unstructured":"Li, C., Yue, Q., Huang, Y.: Two families of nearly optimal codebooks. Des. Codes Crypt. 75(1), 43\u201357 (2015)","journal-title":"Des. Codes Crypt."},{"issue":"9","key":"194_CR18","doi-asserted-by":"crossref","first-page":"5760","DOI":"10.1109\/TIT.2014.2339861","volume":"60","author":"N Li","year":"2014","unstructured":"Li, N., Tang, X., Helleseth, T.: New constructions of quadratic bent functions in pilynomial form. IEEE Trans. Inf. Theory 60(9), 5760\u20135767 (2014)","journal-title":"IEEE Trans. Inf. Theory"},{"key":"194_CR19","first-page":"43","volume":"40","author":"VI Levenshtein","year":"1983","unstructured":"Levenshtein, V.I.: Bounds for packing of metric spaces and some of their applications. Probl. Cybern. 40, 43\u2013110 (1983)","journal-title":"Probl. Cybern."},{"key":"194_CR20","first-page":"63","volume-title":"Welch\u2019s bound and sequence set for code-division muitiple-access systems","author":"JL Massey","year":"1993","unstructured":"Massey, J.L., Mittelholzer, T.: Welch\u2019s bound and sequence set for code-division muitiple-access systems, pp 63\u201378. Springer-verlag, NY, USA (1993)"},{"key":"194_CR21","volume-title":"Bent functions: Fundamentals and Results","author":"S Mesnager","year":"2015","unstructured":"Mesnager, S.: Bent functions: Fundamentals and Results. Springer-Verlag, Heidelberg, Germany (2015)"},{"key":"194_CR22","first-page":"63","volume-title":"Meeting the Welch bound with equality","author":"D Sarwate","year":"1999","unstructured":"Sarwate, D.: Meeting the Welch bound with equality, pp 63\u201379. Proceedings of SETA\u201998, Springer-Verlag, Berlin, Heidelberg (1999)"},{"key":"194_CR23","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1016\/S1063-5203(03)00023-X","volume":"14","author":"JrT Strohmer","year":"2003","unstructured":"Strohmer, Jr. T., Heath, R.W.: Grassmannian frames with applications to coding and communications. Appl. Comput. Harmonic Anal. 14, 257\u2013275 (2003)","journal-title":"Appl. Comput. Harmonic Anal."},{"issue":"12","key":"194_CR24","doi-asserted-by":"crossref","first-page":"5803","DOI":"10.1109\/TIT.2009.2032818","volume":"55","author":"K Schmidt","year":"2009","unstructured":"Schmidt, K.: \u2124 4 $\\mathbb {Z}_{4}$ -valued quadratic forms and quaternary sequence families. IEEE Trans. Inf. Theory 55(12), 5803\u20135810 (2009)","journal-title":"IEEE Trans. Inf. Theory"},{"key":"194_CR25","unstructured":"Sol\u00e9, P., Tokareva, N.: Connections between quaternary and binary bent functions. [Online]. Available: http:\/\/www.eprint.iacr.org\/2009\/544 (2009)"},{"key":"194_CR26","doi-asserted-by":"crossref","unstructured":"Welch, L.: Lower bounds on the maximum cross correlation of signal. IEEE Trans. Inf. Theory IT-20, 397\u2013399 (1974)","DOI":"10.1109\/TIT.1974.1055219"},{"issue":"2","key":"194_CR27","doi-asserted-by":"crossref","first-page":"363","DOI":"10.1016\/0003-4916(89)90322-9","volume":"191","author":"W Wooters","year":"1989","unstructured":"Wooters, W., Fields, B.: Optimal state-determination by mutually unbiased measurements. Ann. Phys. 191(2), 363\u2013381 (1989)","journal-title":"Ann. Phys."},{"issue":"5","key":"194_CR28","doi-asserted-by":"crossref","first-page":"1900","DOI":"10.1109\/TIT.2005.846411","volume":"51","author":"P Xia","year":"2005","unstructured":"Xia, P., Zhou, S., Giannakis, G.B.: Achieving the Welch bound with difference sets. IEEE Trans. Inf. Theory 51(5), 1900\u20131907 (2005)","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"12","key":"194_CR29","doi-asserted-by":"crossref","first-page":"6526","DOI":"10.1109\/TIT.2015.2487451","volume":"61","author":"C Xiang","year":"2015","unstructured":"Xiang, C., Ding, C., Mesnager, S.: Optimal codebooks from binary codebooks meeting the Levenshtein bound. IEEE Trans. Inf. Theory 61(12), 6526\u20136535 (2015)","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"2","key":"194_CR30","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1007\/s10623-011-9549-2","volume":"63","author":"A Zhang","year":"2012","unstructured":"Zhang, A., Feng, K.: Construction of cyclotomic codebooks nearly meeting the Welch bound. Des. Codes Crypt. 63(2), 209\u2013224 (2012)","journal-title":"Des. Codes Crypt."},{"issue":"4","key":"194_CR31","doi-asserted-by":"crossref","first-page":"2507","DOI":"10.1109\/TIT.2011.2176531","volume":"58","author":"A Zhang","year":"2009","unstructured":"Zhang, A., Feng, K.: Two classes of codebooks nearly meeting the Welch bound. IEEE Trans. Inf. Theory 58(4), 2507\u20132511 (2009)","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"11","key":"194_CR32","doi-asserted-by":"crossref","first-page":"7382","DOI":"10.1109\/TIT.2014.2353052","volume":"60","author":"Z Zhou","year":"2014","unstructured":"Zhou, Z., Ding, C., Li, N.: New families of codebooks achieving the Levenshtein bound. IEEE Trans. Inf. Theory 60(11), 7382\u20137387 (2014)","journal-title":"IEEE Trans. Inf. Theory"}],"container-title":["Cryptography and Communications"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s12095-016-0194-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s12095-016-0194-5\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s12095-016-0194-5","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s12095-016-0194-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2017,6,24]],"date-time":"2017-06-24T16:11:51Z","timestamp":1498320711000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s12095-016-0194-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,6,11]]},"references-count":32,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2017,1]]}},"alternative-id":["194"],"URL":"https:\/\/doi.org\/10.1007\/s12095-016-0194-5","relation":{},"ISSN":["1936-2447","1936-2455"],"issn-type":[{"value":"1936-2447","type":"print"},{"value":"1936-2455","type":"electronic"}],"subject":[],"published":{"date-parts":[[2016,6,11]]}}}