{"title":"The Smallest Critical Sets of Latin Squares","authors":"Keith Hermiston","doi":"10.1109/ICSPCS.2018.8631759","DOIUrl":null,"url":null,"abstract":"Latin squares are combinatorial constructions that have found widespread application in communication systems through frequency hopping designs, error correcting codes and encryption algorithms. In this paper, a new, lower bound on the cardinality of the critical sets of all Latin squares of order n is shown to be Ω(n!(n-3)!) where Ω is the summatory prime factorisation function (with multiplicities). The proof draws on the direct product of the symmetric groups Sn and Sn-3. The smallest critical set cardinalities of the new bound align with its known, calculated values and reduce previously proven bounds for n > 8. The proof refutes the long standing Nelder conjecture.","PeriodicalId":179948,"journal":{"name":"2018 12th International Conference on Signal Processing and Communication Systems (ICSPCS)","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2018-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"2018 12th International Conference on Signal Processing and Communication Systems (ICSPCS)","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1109/ICSPCS.2018.8631759","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 1
Abstract
Latin squares are combinatorial constructions that have found widespread application in communication systems through frequency hopping designs, error correcting codes and encryption algorithms. In this paper, a new, lower bound on the cardinality of the critical sets of all Latin squares of order n is shown to be Ω(n!(n-3)!) where Ω is the summatory prime factorisation function (with multiplicities). The proof draws on the direct product of the symmetric groups Sn and Sn-3. The smallest critical set cardinalities of the new bound align with its known, calculated values and reduce previously proven bounds for n > 8. The proof refutes the long standing Nelder conjecture.