{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,6,18]],"date-time":"2025-06-18T04:14:09Z","timestamp":1750220049524,"version":"3.41.0"},"reference-count":8,"publisher":"Association for Computing Machinery (ACM)","issue":"2","license":[{"start":{"date-parts":[[2022,6,1]],"date-time":"2022-06-01T00:00:00Z","timestamp":1654041600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.acm.org\/publications\/policies\/copyright_policy#Background"}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["ACM Commun. Comput. Algebra"],"published-print":{"date-parts":[[2022,6]]},"abstract":"<jats:p>In this work, we explore the relationship between free resolution of some monomial ideals and Generalized Hamming Weights (GHWs) of binary codes. More precisely, we look for a structure smaller than the set of codewords of minimal support that provides us some information about the GHWs. We prove that the first and second generalized Hamming weight of a binary linear code can be computed (by means of a graded free resolution) from a set of monomials associated to a binomial ideal related with the code. Moreover, the remaining weights are bounded by the Betti numbers for that set.<\/jats:p>","DOI":"10.1145\/3572867.3572868","type":"journal-article","created":{"date-parts":[[2022,11,24]],"date-time":"2022-11-24T00:37:32Z","timestamp":1669250252000},"page":"19-24","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":0,"title":["Computing generalized hamming weights of binary linear codes via free resolutions"],"prefix":"10.1145","volume":"56","author":[{"given":"Ignacio","family":"Garc\u00eda-Marco","sequence":"first","affiliation":[{"name":"Universidad de La Laguna, Tenerife, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Irene","family":"M\u00e1rquez-Corbella","sequence":"additional","affiliation":[{"name":"Universidad de La Laguna, Tenerife, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Edgar","family":"Mart\u00ednez-Moro","sequence":"additional","affiliation":[{"name":"University of Valladolid, Valladolid, Castilla, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yuriko","family":"Pitones","sequence":"additional","affiliation":[{"name":"Universidad Aut\u00f3noma Metropolitana, M\u00e9xico D.F., M\u00e9xico"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"320","published-online":{"date-parts":[[2022,11,23]]},"reference":[{"key":"e_1_2_1_1_1","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511662041.008"},{"key":"e_1_2_1_2_1","volume-title":"Gr\u00f6bner bases and combinatorics for binary codes. Applicable Algebra in Engineering, Communication and Computing","author":"Borges-Quintana M.","year":"1996","unstructured":"Borges-Quintana , M. ; Borges-Trenard , M. ; Fitzpatrick , P. ; Mart\u00ednez-Moro , E. Gr\u00f6bner bases and combinatorics for binary codes. Applicable Algebra in Engineering, Communication and Computing 1996 , 19, 1--45. Borges-Quintana, M.; Borges-Trenard, M.; Fitzpatrick, P.; Mart\u00ednez-Moro, E. Gr\u00f6bner bases and combinatorics for binary codes. Applicable Algebra in Engineering, Communication and Computing 1996, 19, 1--45."},{"key":"e_1_2_1_3_1","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-17296-5_15"},{"key":"e_1_2_1_4_1","first-page":"10","article-title":"Free Resolutions and Generalized Hamming Weights of Binary Linear Codes","volume":"2022","author":"Garc\u00eda-Marco I.","year":"2079","unstructured":"Garc\u00eda-Marco , I. ; M\u00e1rquez-Corbella , I. ; Mart\u00ednez-Moro , E. ; Pitones , Y. : Free Resolutions and Generalized Hamming Weights of Binary Linear Codes . Mathematics 2022 , 10 , 2079 . Garc\u00eda-Marco, I.; M\u00e1rquez-Corbella, I.; Mart\u00ednez-Moro, E.; Pitones, Y.: Free Resolutions and Generalized Hamming Weights of Binary Linear Codes. Mathematics 2022, 10, 2079.","journal-title":"Mathematics"},{"key":"e_1_2_1_5_1","doi-asserted-by":"publisher","DOI":"10.1007\/s00200-012-0183-7"},{"key":"e_1_2_1_6_1","doi-asserted-by":"publisher","DOI":"10.3934\/amc.2016003"},{"key":"e_1_2_1_7_1","first-page":"417","volume-title":"Combinatorial commutative algebra","author":"Miller E.","year":"2005","unstructured":"Miller , E. ; Sturmfels , B. Combinatorial commutative algebra ; Vol. 227 , Graduate Texts in Mathematics, Springer-Verlag , New York, 2005 ; pp. xiv+ 417 . Miller, E.; Sturmfels, B. Combinatorial commutative algebra; Vol. 227, Graduate Texts in Mathematics, Springer-Verlag, New York, 2005; pp. xiv+417."},{"key":"e_1_2_1_8_1","volume-title":"the Sage Mathematics Software System (Version 9.5)","author":"The Sage Developers","year":"2022","unstructured":"The Sage Developers . SageMath , the Sage Mathematics Software System (Version 9.5) , 2022 . https:\/\/www.sagemath.org. The Sage Developers. 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