{"pk":64900,"title":"Generalized weights of codes over rings and invariants of monomial ideals","subtitle":null,"abstract":"We develop an algebraic theory of supports for \\(R\\)-linear codes of fixed length, where \\(R\\) is a finite commutative unitary ring. A support naturally induces a notion of generalized weights and allows one to associate a monomial ideal to a code. Our main result states that, under suitable assumptions, the generalized weights of a code can be obtained from the graded Betti numbers of its associated monomial ideal. In the case of \\(\\mathbb{F}_q\\)-linear codes endowed with the Hamming metric, the ideal coincides with the Stanley-Reisner ideal of the matroid associated to the code via its parity-check matrix. In this special setting, we recover the known result that the generalized weights of an \\(\\mathbb{F}_q\\)-linear code can be obtained from the graded Betti numbers of the ideal of the matroid associated to the code. We also study subcodes and codewords of minimal support in a code, proving that a large class of \\(R\\)-linear codes is generated by its codewords of minimal support.\n \nMathematics Subject Classifications: 94B05, 13D02, 13F10\n \nKeywords: Linear codes, codes over rings, supports, generalized weights, monomial ideal of a code, graded Betti numbers, matroid","language":"en","license":{"name":"Creative Commons Attribution 4.0","short_name":"CC BY 4.0","text":"Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.\n\nNo additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.","url":"https://creativecommons.org/licenses/by/4.0"},"keywords":[{"word":"Linear codes"},{"word":"codes over rings"},{"word":"supports"},{"word":"generalized weights"},{"word":"monomial ideal of a code"},{"word":"graded Betti numbers"},{"word":"matroid"}],"section":"Research Articles","is_remote":true,"remote_url":"https://escholarship.org/uc/item/10z7j96g","frozenauthors":[{"first_name":"Elisa","middle_name":"","last_name":"Gorla","name_suffix":"","institution":"Institut de Mathématiques, Université de Neuchâtel, Switzerland","department":""},{"first_name":"Alberto","middle_name":"","last_name":"Ravagnani","name_suffix":"","institution":"Department of Mathematics and Computer Science, Eindhoven University of Technology, the Netherlands","department":""}],"date_submitted":"2023-09-14T07:54:40Z","date_accepted":"2023-09-14T07:54:40Z","date_published":"2023-09-15T07:00:00Z","render_galley":null,"galleys":[{"label":"","type":"pdf","path":"https://journalpub.escholarship.org/combinatorial_theory/article/64900/galley/49710/download/"}]}