{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,23]],"date-time":"2026-08-23T17:26:28Z","timestamp":1787505988952,"version":"build-2736575974"},"reference-count":17,"publisher":"Springer Science and Business Media LLC","issue":"12","license":[{"start":{"date-parts":[[2023,8,3]],"date-time":"2023-08-03T00:00:00Z","timestamp":1691020800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2023,8,3]],"date-time":"2023-08-03T00:00:00Z","timestamp":1691020800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Des. Codes Cryptogr."],"published-print":{"date-parts":[[2023,12]]},"DOI":"10.1007\/s10623-023-01276-8","type":"journal-article","created":{"date-parts":[[2023,8,3]],"date-time":"2023-08-03T07:01:57Z","timestamp":1691046117000},"page":"3857-3879","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["On the fourth weight of generalized Reed\u2013Muller codes"],"prefix":"10.1007","volume":"91","author":[{"given":"Somayyeh","family":"Golalizadeh","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Nasrin","family":"Soltankhah","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2023,8,3]]},"reference":[{"issue":"2","key":"1276_CR1","doi-asserted-by":"publisher","first-page":"313","DOI":"10.3934\/amc.2019021","volume":"13","author":"M Borello","year":"2019","unstructured":"Borello M., Mila O.: Symmetries of weight enumerators and applications to Reed-Muller codes. Adv. Math. Commun. 13(2), 313\u2013328 (2019).","journal-title":"Adv. Math. Commun."},{"key":"1276_CR2","doi-asserted-by":"crossref","unstructured":"Bruen A.A.: Blocking sets and low-weight codewords in the generalized Reed-Muller codes. In: Error Correcting Codes, Finite Geometries and Aryptography. Contemporary Mathematics, vol. 523, pp. 161\u2013164. American Mathematical Society, Providence (2010).","DOI":"10.1090\/conm\/523\/10323"},{"key":"1276_CR3","doi-asserted-by":"publisher","first-page":"42","DOI":"10.1007\/3-540-54303-1_116","volume":"514","author":"C Carlet","year":"1991","unstructured":"Carlet C.: A transformation on Boolean functions, its consequences on some problems related to Reed-Muller codes. Lect. Notes Comput. Sci. 514, 42\u201350 (1991).","journal-title":"Lect. Notes Comput. Sci."},{"issue":"2","key":"1276_CR4","doi-asserted-by":"publisher","first-page":"214","DOI":"10.1006\/ffta.1996.0014","volume":"2","author":"JP Cherdieu","year":"1996","unstructured":"Cherdieu J.P., Rolland R.: On the number of points of some hypersurfaces in $$F^n_q$$. Finite Fields Appl. 2(2), 214\u2013224 (1996).","journal-title":"Finite Fields Appl."},{"key":"1276_CR5","doi-asserted-by":"publisher","first-page":"403","DOI":"10.1016\/S0019-9958(70)90214-7","volume":"16","author":"P Delsarte","year":"1970","unstructured":"Delsarte P., Goethals J.M., MacWilliams F.J.: On generalized Reed-Muller codes and their relatives. Inf. Control 16, 403\u2013442 (1970).","journal-title":"Inf. Control"},{"key":"1276_CR6","unstructured":"Elkies N.D.: Linear codes and algebraic geometry in higher dimensions. Preprint (2006)."},{"key":"1276_CR7","unstructured":"Erickson D., California M.: Institute of technology. Division of physics, astronomy, counting zeros of polynomials over finite fields. CIT Thesis, California Institute of Technology (1974)."},{"issue":"3","key":"1276_CR8","doi-asserted-by":"publisher","first-page":"323","DOI":"10.1007\/s10623-008-9211-9","volume":"48","author":"O Geil","year":"2008","unstructured":"Geil O.: On the second weight of generalized Reed-Muller codes. Des. Codes Cryptogr. 48(3), 323\u2013330 (2008).","journal-title":"Des. Codes Cryptogr."},{"key":"1276_CR9","doi-asserted-by":"publisher","first-page":"189","DOI":"10.1109\/TIT.1968.1054127","volume":"14","author":"T Kasami","year":"1968","unstructured":"Kasami T., Lin S., Peterson W.W.: New generalizations of the Reed-Muller codes, I. Primitive codes. IEEE Trans. Inf. Theory 14, 189\u2013199 (1968).","journal-title":"IEEE Trans. Inf. Theory"},{"key":"1276_CR10","unstructured":"Kasami T., Tokura N., Azumi S.: On the Weight Enumeration of Weights Less than 2.5d of Reed-Muller Codes. Faculty of Engineering Science, Osaka University, Osaka (1974)."},{"issue":"3","key":"1276_CR11","doi-asserted-by":"publisher","first-page":"581","DOI":"10.1016\/j.ffa.2011.12.003","volume":"18","author":"E Leducq","year":"2012","unstructured":"Leducq E.: A new proof of Delsarte, Goethals and Mac Williams theorem on minimal weight codewords of generalized Reed-Muller codes. Finite Fields Appl. 18(3), 581\u2013586 (2012).","journal-title":"Finite Fields Appl."},{"issue":"4","key":"1276_CR12","doi-asserted-by":"publisher","first-page":"241","DOI":"10.1007\/s12095-013-0084-z","volume":"5","author":"E Leducq","year":"2013","unstructured":"Leducq E.: Second weight codewords of generalized Reed-Muller codes. Cryptogr. Commun. 5(4), 241\u2013276 (2013).","journal-title":"Cryptogr. Commun."},{"key":"1276_CR13","doi-asserted-by":"publisher","first-page":"1515","DOI":"10.1016\/j.disc.2015.03.012","volume":"338","author":"E Leducq","year":"2015","unstructured":"Leducq E.: On the third weight of generalized Reed-Muller codes. Discret. Math. 338, 1515\u20131535 (2015).","journal-title":"Discret. Math."},{"key":"1276_CR14","doi-asserted-by":"publisher","first-page":"2447","DOI":"10.1007\/s10623-019-00630-z","volume":"87","author":"S Li","year":"2019","unstructured":"Li S.: On the weight distribution of second order Reed-Muller codes and their relatives. Des. Codes Cryptogr. 87, 2447\u20132460 (2019).","journal-title":"Des. Codes Cryptogr."},{"key":"1276_CR15","first-page":"37","volume":"III","author":"R McEliece","year":"1969","unstructured":"McEliece R.: Quadratic forms over finite fields and second-order Reed-Muller codes, JPL space programs summary. Trans. Inf. Theory III, 37\u201358 (1969).","journal-title":"Trans. Inf. Theory"},{"issue":"1","key":"1276_CR16","doi-asserted-by":"publisher","first-page":"19","DOI":"10.1007\/s12095-009-0014-2","volume":"2","author":"R Rolland","year":"2010","unstructured":"Rolland R.: The second weight of generalized Reed-Muller codes in most cases. Cryptogr. Commun. 2(1), 19\u201340 (2010).","journal-title":"Cryptogr. Commun."},{"issue":"3","key":"1276_CR17","doi-asserted-by":"publisher","first-page":"444","DOI":"10.1016\/j.ffa.2005.11.002","volume":"13","author":"A Sboui","year":"2007","unstructured":"Sboui A.: Second highest number of points of hypersurfaces in $$F^n_q$$. Finite Fields Appl. 13(3), 444\u2013449 (2007).","journal-title":"Finite Fields Appl."}],"container-title":["Designs, Codes and Cryptography"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10623-023-01276-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10623-023-01276-8\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10623-023-01276-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,11,4]],"date-time":"2023-11-04T08:05:00Z","timestamp":1699085100000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10623-023-01276-8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,8,3]]},"references-count":17,"journal-issue":{"issue":"12","published-print":{"date-parts":[[2023,12]]}},"alternative-id":["1276"],"URL":"https:\/\/doi.org\/10.1007\/s10623-023-01276-8","relation":{},"ISSN":["0925-1022","1573-7586"],"issn-type":[{"value":"0925-1022","type":"print"},{"value":"1573-7586","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,8,3]]},"assertion":[{"value":"25 January 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"28 March 2023","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"11 July 2023","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 August 2023","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}