{"title":"环上码的广义权与单项式理想的不变量","authors":"Elisa Gorla, Alberto Ravagnani","doi":"10.5070/c63261989","DOIUrl":null,"url":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.Mathematics Subject Classifications: 94B05, 13D02, 13F10Keywords: Linear codes, codes over rings, supports, generalized weights, monomial ideal of a code, graded Betti numbers, matroid","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2023-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":"{\"title\":\"Generalized weights of codes over rings and invariants of monomial ideals\",\"authors\":\"Elisa Gorla, Alberto Ravagnani\",\"doi\":\"10.5070/c63261989\",\"DOIUrl\":null,\"url\":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.Mathematics Subject Classifications: 94B05, 13D02, 13F10Keywords: Linear codes, codes over rings, supports, generalized weights, monomial ideal of a code, graded Betti numbers, matroid\",\"PeriodicalId\":50230,\"journal\":{\"name\":\"Journal of Combinatorial Theory Series A\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2023-09-14\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"2\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Combinatorial Theory Series A\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.5070/c63261989\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Combinatorial Theory Series A","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.5070/c63261989","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
Generalized weights of codes over rings and invariants of monomial ideals
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.Mathematics Subject Classifications: 94B05, 13D02, 13F10Keywords: Linear codes, codes over rings, supports, generalized weights, monomial ideal of a code, graded Betti numbers, matroid
期刊介绍:
The Journal of Combinatorial Theory publishes original mathematical research concerned with theoretical and physical aspects of the study of finite and discrete structures in all branches of science. Series A is concerned primarily with structures, designs, and applications of combinatorics and is a valuable tool for mathematicians and computer scientists.