On 2-ranks of Steiner triple systems

Abstract

Our main result is an existence and uniqueness theorem for Steiner triple systems which associates to every such system a binary code — called the "carrier" — 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.