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\begin{document}
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\markright{\sc the electronic journal of combinatorics 4 (1997) \#R6}
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\title{Codes, Lattices, and Steiner Systems}
\author{Patrick Sol\'e\thanks{\tt sole@alto.unice.fr},\\
CNRS, I3S,\\ESSI, BP 145,
\\Route des Colles,\\06 903 Sophia Antipolis,\\France}
\date{Submitted: February 16 1996; Accepted: January 31, 1997}
\maketitle
\begin{abstract}
Two classification schemes for Steiner triple systems on 15 points
have been proposed
recently: one based on the binary code spanned by the blocks, the other
on the root system attached to the lattice affinely generated by the blocks.
It is shown here that the two approaches are equivalent.

{\bf 1991 AMS Classification:} Primary: 05B07; Secondary: 11H06, 94B25.
\end{abstract}
\section{Introduction}
It has been known since 1919 \cite{1919} that there are 80 Steiner triple systems on
15 points. Recently, two algebraic invariants have been proposed to
classify them. Let $V$ denote the 35 block vectors $v_i$ of length 15
and hamming weight $3$ of such a system. One can attach to $V$ either
\begin{itemize}
\item the binary linear code $C$ spanned by the vectors of $V$ \cite{TW}
\item the lattice $L:=\{\sum_iz_iv_i:\sum_iz_i =0 \;\&\;z_i\in {\bf Z}\}$ 
\cite{DG}
\end{itemize}
The lattice $L$ has norm $\geq 2$ and  its norm $2$ vectors afford a 
(possibly empty)
root system $R.$
It so happens that exactly 5 non-equivalent codes $C$ and also 5 
non-equivalent root systems $R$ occur and that they induce the same 
partition of the 80 $S(2,3,15)$ in five parts.
We shall provide a conceptual explanation of this experimental fact.

\section{Notations and Definitions}
A Steiner triple system $S(2,3,v)$ is a $2-(v,3,1)$ design.

A binary {\em code} of length $n$ and dimension $k$ is a $k-$dimensional
vector subspace of ${\bf F}_2^n.$ The (Hamming) weight of a vector of
${\bf F}_2^n$ is the number of non-zero coordinates it contains.

An $n-$dimensional {\em lattice} is a discrete $\bf Z-$module of ${\bf R}^n$
which may or may not be of maximal rank ($n.$) The (squared euclidean)
norm of a vector $x$ of ${\bf R}^n$ is $x.x$. The {\em norm} of a 
lattice is the minimum nonzero norm of its elements.
A lattice is {\em integral} if the dot product of any two of its vectors
is an integer.
An integral lattice is called {\em even} (or type II in\cite{SPLAG})
 if the norm of each its vectors is an
even integer.
A {\em root} in an even integral lattice is a vector of norm $2$. A {\em root
system} is the set of all such vectors in an even lattice.
\section{Explanation}
 Let $C_e$ denote the following subcode
$$C_e:=\{\sum_iz_iv_i:\sum_iz_i =0 \;\&\;z_i\in {\bf F}_2\}$$
of $C.$ Recall that construction $A$ of \cite{SPLAG} (here with a different
normalization) associates to a binary 
code $D$ the lattice

$$A(D):=D+2{\bf Z}^n.$$

{\th The code $C_e$ is the even weight subcode of $C$ and
 $$L\subseteq A(C_e).$$}
 
 \pr The second assertion is immediate from the definition of $C_e.$
 The first assertion comes from the fact that
  the sum of coordinates of a typical vector of $ L$ is
 $$\sum_j(\sum_iz_iv_i)_j=\sum_iz_i(\sum_j(v_i)_j)\equiv 0 \,(mod \, 2).$$
 This shows inclusion of $C_e$ into the even weight subcode of $C.$
  Equality comes from the fact that $C_e$
 is generated by $v_1+v_i,\; i =2,\dots, 15,$ which yields the direct
 sum
 $$C={\bf F}_2v_1{\oplus}C_e.$$
 \qed
 
 {\bf Remark:} $L\neq A(C_e)$ for $2v_1\in A(C_e)$ but $2v_1$ is not in $L.$
 While $A(C_e)$ is of maximal rank, $L$ is not.
 To make this remark more precise, we introduce an auxilliary lattice. Let
 $e_i,\,i =1,\dots,15$ denote the canonical basis (i.e. the 15 vectors of shape
 $10^{14}$ ) and call $k$ the dimension of $C_e.$
 Let $L_k$ denote the $\bf Z$-span of the vectors $2e_i, \; i =k+1,\dots, 
 n.$
 {\th  The lattice $L$ is obtained from $A(C_e)$ by successive projections onto
 a vector space:
 $$A(C_e)=2{\bf Z}v_1\oplus L\oplus L_k .$$
 Therefore the root system $R$ depends solely on $C.$}
 
 \pr Let $$L':=\{\sum_iz_iv_i:\sum_iz_i =0\,(mod 2) \;\&\;z_i\in {\bf Z}\}.$$
 It is easy to see, using explicit projectors that 
 $$L'=2{\bf Z}v_1\oplus L.$$
 Furthermore, from the generating matrix for construction $A$ \cite[p.183]{SPLAG}
 we see that 
 $$A(C_e)=L'\oplus L_k.$$
 Combining the last two equations we are done.\qed
 
 We can relate the root system $R$ to the code  $C.$
 {\th The root system $R$ consists of vectors of the shape $(\pm 1)^20^{13}$
  supported by weight $2$ codewords in $C.$}
 
 \pr 
 From Theorem 1 it follows that the vectors of norm $2$ in $L$ are in $A(C_e).$
 It is known that the vectors of norm $2$ of $A(C_e)$ comprise
 suitably signed versions of the vectors of  weight 2 of $C_e,$ i.e. of the
 vectors of  weight 2 of $C.$
 \qed
 
 \section{Conclusion}
 From the preceding results it transpires that the lattice depends
 solely on the code and therefore, by combining with the results in \cite{A,TW},
 since the code depends solely on its dimension, solely on the 2-rank
 of the considered STS. We leave to the interested reader the
 explicit determination of root systems and lattices involved.

 \section{Acknowledgements} We thank Ed Assmus, Michel Deza, and Vladimir
 Tonchev for sending us their preprints and Chris Charnes, Slava Grishukhin for helpful
 discussions. We thank the Mathematics Department of Macquarie University
 for its hospitality.
\begin{thebibliography}{99}
\bibitem{A} E. F. Assmuss, jr. On 2-ranks of Steiner Triple Systems,
Electronic Journal of Combinatorics, 2 (1995), paper R9.
\bibitem{SPLAG}J.H. Conway, N.J.A. Sloane, {\em Sphere Packings Lattices and Groups},
second edition, Springer Verlag (1993).
\bibitem{DG}M. Deza,V. Grishukhin, Once More about 80 Steiner triple systems on
15 points,  LIENS research report 95-8.
\bibitem{TW}V.D. Tonchev, R.S. Weishaar, Steiner Systems of order 15 and their codes,
J. of Stat. Plann. and Inf. submitted (1995).
\bibitem{1919}H. S. White, F.N. Cole, L. D. Cummings, Complete 
Classification of the triad systems on fifteen elements, Mem. Nat. Acad. 
Sc. USA 14, second memoir (1919) 1-89.
\end{thebibliography}
\end{document}