\magnification=1440
\font\bigtenrm=cmr10 scaled\magstep4

Abstract for E. F. Assmus, Jr. On 2-ranks of Steiner triple systems

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 {\it
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  {\it 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. % This part of the paper can be viewed as a
%constructive elaboration of a program begun by Teirlinck and
%brought to what was then seen as a definitive end by Doyen,
%Hubaut and Vandensavel almost twenty years ago.
 
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.

\bye	

.