{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,24]],"date-time":"2026-01-24T17:40:09Z","timestamp":1769276409636,"version":"3.49.0"},"reference-count":25,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2023,12,7]],"date-time":"2023-12-07T00:00:00Z","timestamp":1701907200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2023,12,7]],"date-time":"2023-12-07T00:00:00Z","timestamp":1701907200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100003725","name":"National Research Foundation of Korea","doi-asserted-by":"publisher","award":["NRF-2020R1A6A3A13065516,2019R1A6A1A11051177"],"award-info":[{"award-number":["NRF-2020R1A6A3A13065516,2019R1A6A1A11051177"]}],"id":[{"id":"10.13039\/501100003725","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Cryptogr. Commun."],"published-print":{"date-parts":[[2024,5]]},"DOI":"10.1007\/s12095-023-00686-8","type":"journal-article","created":{"date-parts":[[2023,12,7]],"date-time":"2023-12-07T06:02:07Z","timestamp":1701928927000},"page":"579-600","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["The Griesmer codes of Belov type and optimal quaternary codes via multi-variable functions"],"prefix":"10.1007","volume":"16","author":[{"given":"Jong Yoon","family":"Hyun","sequence":"first","affiliation":[]},{"given":"Nayoung","family":"Han","sequence":"additional","affiliation":[]},{"given":"Yoonjin","family":"Lee","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,12,7]]},"reference":[{"key":"686_CR1","unstructured":"Aydin, N., Asamov, T.: Database of $$\\mathbb{Z}_4$$ codes [online], http:\/\/quantumcodes.info\/Z4, Apr. 15, 2022"},{"key":"686_CR2","first-page":"100","volume":"182","author":"BI Belov","year":"1972","unstructured":"Belov, B.I.: A conjecture on the Griesmer boundary (Russian) optimization methods and their applications, All-Union Summer Sem. Khakusy, Lake Baikal 182, 100\u2013106 (1972)","journal-title":"Khakusy, Lake Baikal"},{"issue":"11","key":"686_CR3","doi-asserted-by":"publisher","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. Inf. Theory 49(11), 2834\u20132843 (2003)","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"8","key":"686_CR4","doi-asserted-by":"publisher","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-$$\\mathbb{Z} _4$$-linear codes. IEEE Trans. Inf. Theory 49(8), 2028\u20132034 (2003)","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"1","key":"686_CR5","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s10623-007-9136-8","volume":"48","author":"E Byrne","year":"2008","unstructured":"Byrne, E., Greferath, M., Honold, T.: Ring geometries, two-weight codes and strongly regular graphs. Des. Codes Cryptogr. 48(1), 1\u201316 (2008)","journal-title":"Des. Codes Cryptogr."},{"issue":"2","key":"686_CR6","doi-asserted-by":"publisher","first-page":"365","DOI":"10.1112\/jlms\/s2-26.2.365","volume":"26","author":"R Calderbank","year":"1982","unstructured":"Calderbank, R.: On uniformly packed $$[n, n-k, 4]$$ codes over $$GF(q)$$ and a class of caps in $$\\operatorname{PG}(k-1, q)$$. J. London Math. Soc. (2) 26(2), 365\u2013384 (1982)","journal-title":"J. London Math. Soc. (2)"},{"issue":"2","key":"686_CR7","doi-asserted-by":"publisher","first-page":"97","DOI":"10.1112\/blms\/18.2.97","volume":"18","author":"R Calderbank","year":"1986","unstructured":"Calderbank, R., Kantor, W.M.: The geometry of two-weight codes. Bull. Lond. Math. Soc. 365-384. 18(2), 97\u2013122 (1986)","journal-title":"Bull. Lond. Math. Soc. 365-384."},{"key":"686_CR8","doi-asserted-by":"publisher","first-page":"47","DOI":"10.1016\/0012-365X(72)90024-6","volume":"3","author":"P Delsarte","year":"1972","unstructured":"Delsarte, P.: Weights of linear codes and strongly regular normed spaces. Discrete Math. 3, 47\u201364 (1972)","journal-title":"Discrete Math."},{"issue":"2","key":"686_CR9","doi-asserted-by":"publisher","first-page":"301","DOI":"10.1109\/18.312154","volume":"40","author":"R Hammons","year":"1994","unstructured":"Hammons, R., Kumar, V.P., Calderbank, A.R., Sloane, N.J.A., Sol\u00e9, P.: The $$\\mathbb{Z} _4$$-linearity of Kerdock, Preparata, Goethals and related codes. IEEE Trans. Inf. Theory 40(2), 301\u2013319 (1994)","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"2","key":"686_CR10","doi-asserted-by":"publisher","first-page":"798","DOI":"10.1109\/18.661524","volume":"44","author":"X Hou","year":"1998","unstructured":"Hou, X., Lahtonen, J., Koponen, S.: The Reed-Muller code R(r, m) is not $$\\mathbb{Z} _4$$-linear for $$3 < r < m-2$$. IEEE Trans. Inf. Theory 44(2), 798\u2013799 (1998)","journal-title":"IEEE Trans. Inf. Theory"},{"key":"686_CR11","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511807077","volume-title":"Fundamentals of Error Correcting codes","author":"WC Huffman","year":"2003","unstructured":"Huffman, W.C., Pless, V.: Fundamentals of Error Correcting codes. Cambridge University Press, Cambridge (2003)"},{"issue":"11","key":"686_CR12","doi-asserted-by":"publisher","first-page":"6762","DOI":"10.1109\/TIT.2020.2993179","volume":"66","author":"JY Hyun","year":"2020","unstructured":"Hyun, J.Y., Lee, J., Lee, Y.: Infinite families of optimal linear codes constructed from simplicial complexes. IEEE Trans. Inf. Theory 66(11), 6762\u20136773 (2020)","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"2","key":"686_CR13","doi-asserted-by":"publisher","first-page":"227","DOI":"10.3934\/amc.2020055","volume":"15","author":"JY Hyun","year":"2021","unstructured":"Hyun, J.Y., Kim, B., Na, M.: Construction of minimal linear codes from multi-variable functions. Adv. Math. Commun. 15(2), 227\u2013240 (2021)","journal-title":"Adv. Math. Commun."},{"issue":"4","key":"686_CR14","first-page":"78","volume":"7","author":"DS Krotov","year":"2000","unstructured":"Krotov, D.S.: $$\\mathbb{Z} _4$$-linear perfect codes. Diskretn. Anal. Issled. Oper. Ser. 1 7(4), 78\u201390 (2000)","journal-title":"Diskretn. Anal. Issled. Oper. Ser. 1"},{"key":"686_CR15","volume-title":"The Theory of Error-Correcting Codes","author":"FJ MacWilliams","year":"1977","unstructured":"MacWilliams, F.J., Sloane, N.J.A.: The Theory of Error-Correcting Codes. North-Holland, Amsterdam, The Netherlands (1977)"},{"issue":"9","key":"686_CR16","doi-asserted-by":"publisher","first-page":"6043","DOI":"10.1109\/TIT.2011.2119465","volume":"57","author":"J Pernas","year":"2011","unstructured":"Pernas, J., Pujol, J., Villanueva, M.: Classification of some families of quaternary Reed-Muller codes. IEEE Trans. Inform. Theory 57(9), 6043\u20136051 (2011)","journal-title":"IEEE Trans. Inform. Theory"},{"issue":"1","key":"686_CR17","doi-asserted-by":"publisher","first-page":"99","DOI":"10.1109\/TIT.2008.2008143","volume":"55","author":"J Pujol","year":"2009","unstructured":"Pujol, J., Rif\u00e0, J., Solov\u00e1, F.I.: Construction of $$\\mathbb{Z} _4$$-linear Reed-Muller codes. IEEE Trans. Inf. Theory 55(1), 99\u2013104 (2009)","journal-title":"IEEE Trans. Inf. Theory"},{"key":"686_CR18","volume-title":"Codes and rings","author":"M Shi","year":"2017","unstructured":"Shi, M., Alahmadi, A., Sol\u00e9, P.: Codes and rings. Academic Press, London, Theory and practice (2017)"},{"issue":"12","key":"686_CR19","doi-asserted-by":"publisher","first-page":"7769","DOI":"10.1109\/TIT.2021.3114636","volume":"67","author":"Minjia Shi","year":"2021","unstructured":"Shi, Minjia, Honold, Thomas, Sol\u00e9, Patrick, Qiu, Yunzhen, Rongsheng, Wu., Sepasdar, Zahra: The Geometry of Two-Weight Codes Over $$\\mathbb{Z} _{p^m}$$. IEEE Trans. Inf. Theory 67(12), 7769\u20137781 (2021)","journal-title":"IEEE Trans. Inf. Theory"},{"key":"686_CR20","doi-asserted-by":"publisher","first-page":"86","DOI":"10.1016\/j.ffa.2016.11.016","volume":"45","author":"Minjia Shi","year":"2017","unstructured":"Shi, Minjia, Qian, Liqin, Sok, Lin, Aydin, Nuh, Sol\u00e9, Patrick: On constacyclic codes over $$\\mathbb{Z} _4 [u] \/ \\langle u^2 - 1 \\rangle $$ and their Gray images. Finite Fields Appl. 45, 86\u201395 (2017)","journal-title":"Finite Fields Appl."},{"issue":"6","key":"686_CR21","doi-asserted-by":"publisher","first-page":"1201","DOI":"10.1007\/s10623-017-0390-0","volume":"86","author":"M Shi","year":"2018","unstructured":"Shi, M., Sepasdar, Z., Alahmadi, A., Sol\u00e9, P.: On two-weight $$\\mathbb{Z} _{2^k}$$-codes. Des. Codes Cryptogr. 86(6), 1201\u20131209 (2018)","journal-title":"Des. Codes Cryptogr."},{"issue":"2","key":"686_CR22","doi-asserted-by":"publisher","first-page":"170","DOI":"10.1016\/S0019-9958(65)90080-X","volume":"8","author":"G Solomon","year":"1965","unstructured":"Solomon, G., Stiffler, J.J.: Algebraically punctured cyclic codes. Inf. Control 8(2), 170\u2013179 (1965)","journal-title":"Inf. Control"},{"issue":"4","key":"686_CR23","doi-asserted-by":"publisher","first-page":"795","DOI":"10.1007\/s11424-014-2188-8","volume":"27","author":"M Shi","year":"2014","unstructured":"Shi, M., Wang, Y.: Optimal binary codes from one-Lee weight codes and two-Lee weight projective codes over $$\\mathbb{Z} _4$$. J. Syst. Sci. Complex. 27(4), 795\u2013810 (2014)","journal-title":"J. Syst. Sci. Complex."},{"issue":"12","key":"686_CR24","doi-asserted-by":"publisher","first-page":"2493","DOI":"10.1007\/s10623-020-00796-x","volume":"88","author":"M Shi","year":"2020","unstructured":"Shi, M., Xuan, W., Sol\u00e9, P.: Two families of two-weight codes over $$\\mathbb{Z} _4$$. Des. Codes Cryptogr. 88(12), 2493\u20132505 (2020)","journal-title":"Des. Codes Cryptogr."},{"issue":"1","key":"686_CR25","doi-asserted-by":"publisher","first-page":"20","DOI":"10.1109\/LCOMM.2019.2950401","volume":"24","author":"X Zhu","year":"2020","unstructured":"Zhu, X., Wu, Y., Yue, Q.: New Quaternary Codes Derived from Posets of the Disjoint Union of Two Chains. IEEE Comm. Lett. 24(1), 20\u201324 (2020)","journal-title":"IEEE Comm. Lett."}],"container-title":["Cryptography and Communications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12095-023-00686-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s12095-023-00686-8\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12095-023-00686-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,4,23]],"date-time":"2024-04-23T09:13:33Z","timestamp":1713863613000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s12095-023-00686-8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,12,7]]},"references-count":25,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2024,5]]}},"alternative-id":["686"],"URL":"https:\/\/doi.org\/10.1007\/s12095-023-00686-8","relation":{},"ISSN":["1936-2447","1936-2455"],"issn-type":[{"value":"1936-2447","type":"print"},{"value":"1936-2455","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,12,7]]},"assertion":[{"value":"18 June 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"14 November 2023","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"7 December 2023","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare no competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}]}}