{"title":"FPT-Algorithms for the (ell) -Matchoid Problem with a Coverage Objective","authors":"Chien-Chung Huang, Justin Ward","doi":"10.1137/21m1442267","DOIUrl":"https://doi.org/10.1137/21m1442267","url":null,"abstract":"","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84945186","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Anina Gruica, A. Ravagnani, John Sheekey, Ferdinando Zullo
We investigate two fundamental questions intersecting coding theory and combinatorial geometry, with emphasis on their connections. These are the problem of computing the asymptotic density of MRD codes in the rank metric, and the Critical Problem for combinatorial geometries by Crapo and Rota. Using methods from semifield theory, we derive two lower bounds for the density function of full-rank, square MRD codes. The first bound is sharp when the matrix size is a prime number and the underlying field is sufficiently large, while the second bound applies to the binary field. We then take a new look at the Critical Problem for combinatorial geometries, approaching it from a qualitative, often asymptotic, viewpoint. We illustrate the connection between this very classical problem and that of computing the asymptotic density of MRD codes. Finally, we study the asymptotic density of some special families of codes in the rank metric, including the symmetric, alternating and Hermitian ones. In particular, we show that the optimal codes in these three contexts are sparse.
{"title":"Rank-Metric Codes, Semifields, and the Average Critical Problem","authors":"Anina Gruica, A. Ravagnani, John Sheekey, Ferdinando Zullo","doi":"10.1137/22m1486893","DOIUrl":"https://doi.org/10.1137/22m1486893","url":null,"abstract":"We investigate two fundamental questions intersecting coding theory and combinatorial geometry, with emphasis on their connections. These are the problem of computing the asymptotic density of MRD codes in the rank metric, and the Critical Problem for combinatorial geometries by Crapo and Rota. Using methods from semifield theory, we derive two lower bounds for the density function of full-rank, square MRD codes. The first bound is sharp when the matrix size is a prime number and the underlying field is sufficiently large, while the second bound applies to the binary field. We then take a new look at the Critical Problem for combinatorial geometries, approaching it from a qualitative, often asymptotic, viewpoint. We illustrate the connection between this very classical problem and that of computing the asymptotic density of MRD codes. Finally, we study the asymptotic density of some special families of codes in the rank metric, including the symmetric, alternating and Hermitian ones. In particular, we show that the optimal codes in these three contexts are sparse.","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74703849","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pierre Bergé, Wassim Bouaziz, Arpad Rimmel, J. Tomasik
{"title":"On the Parameterized Complexity of Counting Small-Sized Minimum (boldsymbol{(S,T)})-Cuts","authors":"Pierre Bergé, Wassim Bouaziz, Arpad Rimmel, J. Tomasik","doi":"10.1137/21m1398203","DOIUrl":"https://doi.org/10.1137/21m1398203","url":null,"abstract":"","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77238233","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Colorful Matchings","authors":"A. Arman, V. Rödl, M. Sales","doi":"10.1137/21m139997x","DOIUrl":"https://doi.org/10.1137/21m139997x","url":null,"abstract":"","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77778001","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"(boldsymbol{H})-Games Played on Vertex Sets of Random Graphs","authors":"Gal Kronenberg, Adva Mond, A. Naor","doi":"10.1137/20m1375991","DOIUrl":"https://doi.org/10.1137/20m1375991","url":null,"abstract":"","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75506236","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On Density of (boldsymbol{mathbb{Z}_3}) -Flow-Critical Graphs","authors":"Z. Dvořák, B. Mohar","doi":"10.1137/22m1496529","DOIUrl":"https://doi.org/10.1137/22m1496529","url":null,"abstract":"","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76129444","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
N. Peyerimhoff, M. Roth, Johannes Schmitt, J. Stix, A. Vdovina, Philip Wellnitz
{"title":"Parameterized Counting and Cayley Graph Expanders","authors":"N. Peyerimhoff, M. Roth, Johannes Schmitt, J. Stix, A. Vdovina, Philip Wellnitz","doi":"10.1137/22m1479804","DOIUrl":"https://doi.org/10.1137/22m1479804","url":null,"abstract":"","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73616417","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-04-06DOI: 10.1016/j.disc.2017.08.022
Louis Anthony Agong, Carmen Amarra, IV JohnS.Caughman, Ari J. Herman, Tai Terada
{"title":"On the girth and diameter of generalized Johnson graphs","authors":"Louis Anthony Agong, Carmen Amarra, IV JohnS.Caughman, Ari J. Herman, Tai Terada","doi":"10.1016/j.disc.2017.08.022","DOIUrl":"https://doi.org/10.1016/j.disc.2017.08.022","url":null,"abstract":"","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90332329","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-04-01DOI: 10.48550/arXiv.2206.01526
D. Kolupaev, A. Kupavskii
{"title":"Erdős Matching Conjecture for almost perfect matchings","authors":"D. Kolupaev, A. Kupavskii","doi":"10.48550/arXiv.2206.01526","DOIUrl":"https://doi.org/10.48550/arXiv.2206.01526","url":null,"abstract":"","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76438784","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-03-25DOI: 10.48550/arXiv.2303.14579
T. Lidbetter
Let $S$ be a finite subset of $mathbb{Z}^n$. A vector sequence $(mathbf{z}_i)$ is an $S$-walk if and only if $mathbf{z}_{i+1} - mathbf{z}_i$ is an element of $S$ for all $i$. Gerver and Ramsey showed in 1979 that for $Ssubset mathbb{Z}^3$ there exists an infinite $S$-walk in which no $5^{11} + 1=48{small,}828{small,}126$ points are collinear. Here, we use the same general approach, but with the aid of a computer search, to improve the bound to $189$.
{"title":"Improved Bound for the Gerver-Ramsey Collinearity Problem","authors":"T. Lidbetter","doi":"10.48550/arXiv.2303.14579","DOIUrl":"https://doi.org/10.48550/arXiv.2303.14579","url":null,"abstract":"Let $S$ be a finite subset of $mathbb{Z}^n$. A vector sequence $(mathbf{z}_i)$ is an $S$-walk if and only if $mathbf{z}_{i+1} - mathbf{z}_i$ is an element of $S$ for all $i$. Gerver and Ramsey showed in 1979 that for $Ssubset mathbb{Z}^3$ there exists an infinite $S$-walk in which no $5^{11} + 1=48{small,}828{small,}126$ points are collinear. Here, we use the same general approach, but with the aid of a computer search, to improve the bound to $189$.","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82115736","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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