{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,1]],"date-time":"2026-04-01T14:19:24Z","timestamp":1775053164776,"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>Our main result is an existence and uniqueness theorem for Steiner triple systems which associates to every such system a binary code \u2014 called the \"carrier\" \u2014 which depends only on the order of the system and its 2-rank.  When the Steiner triple system is   of 2-rank less than the number of points of the system, the carrier organizes all the information necessary to construct directly  all systems of the given order and $2$-rank from Steiner triple systems of a specified  smaller order. The carriers are an easily understood, two-parameter family of binary codes related to the  Hamming codes.  We also discuss Steiner quadruple systems and prove an analogous existence and uniqueness theorem; in this case the binary code (corresponding to the carrier in the triple system case) is the dual of the code obtained from a first-order Reed-Muller code by repeating it a certain specified number of times. Some particularly intriguing possible enumerations and some general open problems are discussed.  We also present applications of this coding-theoretic classification to the theory of triple and quadruple systems giving, for example, a direct proof of the fact that all triple systems are derived provided those of full 2-rank are and showing that whenever there are resolvable quadruple systems on $u$ and on $v$ points there is a resolvable quadruple system on $uv$ points. The methods used in both the classification and the applications make it abundantly clear why the number of triple and quadruple systems grows in such a staggering way and why a triple system that extends to a quadruple system has, generally, many such extensions.<\/jats:p>","DOI":"10.37236\/1203","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T01:26:09Z","timestamp":1578705969000},"source":"Crossref","is-referenced-by-count":23,"title":["On 2-ranks of Steiner triple systems"],"prefix":"10.37236","volume":"2","author":[{"given":"E. F.","family":"Assmus Jr.","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[1995,4,17]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v2i1r9\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v2i1r9\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T09:32:11Z","timestamp":1579339931000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v2i1r9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1995,4,17]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[1995,1,1]]}},"URL":"https:\/\/doi.org\/10.37236\/1203","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[1995,4,17]]},"article-number":"R9"}}