On 2-ranks of Steiner triple systems (Q1804181)

Summary: 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 necessry 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.