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dc.contributor.authorBeane, Maria Elizabethen_US
dc.date.accessioned2014-03-14T20:36:33Z
dc.date.available2014-03-14T20:36:33Z
dc.date.issued2011-04-28en_US
dc.identifier.otheretd-05122011-153554en_US
dc.identifier.urihttp://hdl.handle.net/10919/32676
dc.description.abstractS(5,8,24) is one of the largest known Steiner systems and connects combinatorial designs, error-correcting codes, finite simple groups, and sphere packings in a truly remarkable way. This thesis discusses the underlying structure of S(5,8,24), its construction via the (24,12) Golay code, as well its automorphism group, which is the Mathieu group M24, a member of the sporadic simple groups. Particular attention is paid to the calculation of the size of automorphism groups of Steiner systems using the Orbit-Stabilizer Theorem. We conclude with a section on the sphere packing problem and elaborate on how the 8-sets of S(5,8,24) can be used to form Leech's Lattice, which Leech used to create the densest known sphere packing in 24-dimensions. The appendix contains code written for Matlab which has the ability to construct the octads of S(5,8,24), permute the elements to obtain isomorphic S(5,8,24) systems, and search for certain subsets of elements within the octads.en_US
dc.publisherVirginia Techen_US
dc.relation.haspartBeane_ME_T_2011.pdfen_US
dc.rightsI hereby certify that, if appropriate, I have obtained and attached hereto a written permission statement from the owner(s) of each third party copyrighted matter to be included in my thesis, dissertation, or project report, allowing distribution as specified below. I certify that the version I submitted is the same as that approved by my advisory committee. I hereby grant to Virginia Tech or its agents the non-exclusive license to archive and make accessible, under the conditions specified below, my thesis, dissertation, or project report in whole or in part in all forms of media, now or hereafter known. I retain all other ownership rights to the copyright of the thesis, dissertation or project report. I also retain the right to use in future works (such as articles or books) all or part of this thesis, dissertation, or project report.en_US
dc.subjectSteiner Systemsen_US
dc.subjectError-Correcting Codesen_US
dc.subjectMathieu Groupsen_US
dc.subjectSphere Packingsen_US
dc.titleAn Introduction to S(5,8,24)en_US
dc.typeThesisen_US
dc.contributor.departmentMathematicsen_US
dc.description.degreeMaster of Scienceen_US
thesis.degree.nameMaster of Scienceen_US
thesis.degree.levelmastersen_US
thesis.degree.grantorVirginia Polytechnic Institute and State Universityen_US
thesis.degree.disciplineMathematicsen_US
dc.contributor.committeechairBrown, Ezra A.en_US
dc.contributor.committeememberRossi, John F.en_US
dc.contributor.committeememberShimozono, Mark M.en_US
dc.identifier.sourceurlhttp://scholar.lib.vt.edu/theses/available/etd-05122011-153554/en_US
dc.date.sdate2011-05-12en_US
dc.date.rdate2011-06-01
dc.date.adate2011-06-01en_US


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