{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:44Z","timestamp":1753893824711,"version":"3.41.2"},"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>Let $K_q(n,R)$ denote the minimal cardinality of a $q$-ary code of length $n$ and covering radius $R$. Recently the authors gave a new proof of a classical lower bound of Rodemich on $K_q(n,n-2)$ by the use of partition matrices and their transversals. In this paper we show that, in contrast to Rodemich's original proof, the method generalizes to lower-bound $K_q(n,n-k)$ for any $k&gt;2$. The approach is best-understood in terms of a game where a winning strategy for one of the players implies the non-existence of a code. This proves to be by far the most efficient method presently known to lower-bound $K_q(n,R)$ for large $R$ (i.e. small $k$). One instance: the trivial sphere-covering bound $K_{12}(7,3)\\geq 729$, the previously best bound $K_{12}(7,3)\\geq 732$ and the new bound $K_{12}(7,3)\\geq 878$.<\/jats:p>","DOI":"10.37236\/222","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T04:24:12Z","timestamp":1578716652000},"source":"Crossref","is-referenced-by-count":0,"title":["Lower Bounds for $q$-ary Codes with Large Covering Radius"],"prefix":"10.37236","volume":"16","author":[{"given":"Wolfgang","family":"Haas","sequence":"first","affiliation":[]},{"given":"Immanuel","family":"Halupczok","sequence":"additional","affiliation":[]},{"given":"Jan-Christoph","family":"Schlage-Puchta","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2009,11,7]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v16i1r133\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v16i1r133\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T02:39:57Z","timestamp":1579315197000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v16i1r133"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,11,7]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2009,1,7]]}},"URL":"https:\/\/doi.org\/10.37236\/222","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2009,11,7]]},"article-number":"R133"}}