{"title":"q-matroid 的循环平面","authors":"Gianira N. Alfarano, Eimear Byrne","doi":"10.1007/s10801-024-01321-2","DOIUrl":null,"url":null,"abstract":"<p>In this paper we develop the theory of cyclic flats of <i>q</i>-matroids. We show that the cyclic flats, together with their ranks, uniquely determine a <i>q</i>-matroid and hence derive a new <i>q</i>-cryptomorphism. We introduce the notion of <span>\\(\\mathbb {F}_{q^m}\\)</span>-independence of an <span>\\(\\mathbb {F}_q\\)</span>-subspace of <span>\\(\\mathbb {F}_q^n\\)</span> and we show that <i>q</i>-matroids generalize this concept, in the same way that matroids generalize the notion of linear independence of vectors over a given field.\n</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The cyclic flats of a q-matroid\",\"authors\":\"Gianira N. Alfarano, Eimear Byrne\",\"doi\":\"10.1007/s10801-024-01321-2\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>In this paper we develop the theory of cyclic flats of <i>q</i>-matroids. We show that the cyclic flats, together with their ranks, uniquely determine a <i>q</i>-matroid and hence derive a new <i>q</i>-cryptomorphism. We introduce the notion of <span>\\\\(\\\\mathbb {F}_{q^m}\\\\)</span>-independence of an <span>\\\\(\\\\mathbb {F}_q\\\\)</span>-subspace of <span>\\\\(\\\\mathbb {F}_q^n\\\\)</span> and we show that <i>q</i>-matroids generalize this concept, in the same way that matroids generalize the notion of linear independence of vectors over a given field.\\n</p>\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-05-09\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s10801-024-01321-2\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10801-024-01321-2","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
摘要
在本文中，我们发展了 q-matroids的循环平面理论。我们证明了循环平面连同它们的等级唯一地决定了一个 q-matroid，并由此推导出一个新的 q-密码同构。我们引入了 \(\mathbb {F}_{q^m}\)-independence of an \(\mathbb {F}_q\)-subspace of \(\mathbb {F}_q^n\)子空间的 \(\mathbb {F}_{q^m}\)-independence 概念，并证明了 q-matroids 对这个概念的概括，就像 matroids 对给定域上向量的线性独立性概念的概括一样。
In this paper we develop the theory of cyclic flats of q-matroids. We show that the cyclic flats, together with their ranks, uniquely determine a q-matroid and hence derive a new q-cryptomorphism. We introduce the notion of \(\mathbb {F}_{q^m}\)-independence of an \(\mathbb {F}_q\)-subspace of \(\mathbb {F}_q^n\) and we show that q-matroids generalize this concept, in the same way that matroids generalize the notion of linear independence of vectors over a given field.
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