{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T12:59:26Z","timestamp":1772283566334,"version":"3.50.1"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>All linear codes of length $100$ over a field $F$ which admit the Higman-Sims simple group HS in its rank $3$ representation are determined. By group representation theory it is proved that they can all be understood as submodules of the permutation module $F\\Omega$ where $\\Omega$ denotes the vertex set of the Higman-Sims graph. This module is semisimple if $\\mathrm{char} F\\neq 2,5$ and absolutely indecomposable otherwise. Also if $\\mathrm{char} F \\in \\{2, 5\\}$ the submodule lattice is determined explicitly. The binary case $F = \\mathbb{F}_2$ is studied in detail under coding theoretic aspects. The HS-orbits in the subcodes of dimension $\\leq 23$ are computed explicitly and so also the weight enumerators are obtained. The weight enumerators of the dual codes are determined by MacWilliams transformation. Two fundamental methods are used: Let $v$ be the endomorphism determined by an adjacency matrix. Then in $H_{22} = \\mathrm{Im} v $ the HS-orbits are determined as $v$-images of certain low weight vectors in $F\\Omega$ which carry some special graph configurations. The second method consists in using the fact that $H_{23}\/H_{21}$ is a Klein four group under addition, if $H_{23}$ denotes the code generated by $H_{22}$ and a \"Higman vector\" $x(m)$ of weight 50 associated to a heptad $m$ in the shortened Golay code $G_{22}$, and $H_{21}$ denotes the doubly even subcode of $H_{22}\\leq H_{78} = {H_{22}}^\\perp$. Using the mentioned observation about $H_{23}\/H_{21}$ and the results on the HS-orbits in $H_{23}$ a model of G. Higman's geometry is constructed, which leads to a direct geometric proof that G. Higman's simple group is isomorphic to HS. Finally, it is shown that almost all maximal subgroups of the Higman-Sims group can be understood as stabilizers in HS of codewords in $H_{23}$.<\/jats:p>","DOI":"10.37236\/4267","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T19:33:02Z","timestamp":1578684782000},"source":"Crossref","is-referenced-by-count":5,"title":["On the Codes Related to the Higman-Sims Graph"],"prefix":"10.37236","volume":"22","author":[{"given":"Wolfgang","family":"Knapp","sequence":"first","affiliation":[]},{"given":"Hans-J\u00f6rg","family":"Schaeffer","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2015,1,27]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v22i1p19\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v22i1p19\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T05:37:56Z","timestamp":1579239476000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v22i1p19"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,1,27]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2015,1,2]]}},"URL":"https:\/\/doi.org\/10.37236\/4267","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,1,27]]},"article-number":"P1.19"}}