1. Introduction
A
design
is an incidence structure with
points,
points on each block and any subset of
points is contained in exactly
blocks, where
. the number of blocks is
and the number of blocks on a point is
.
The design
is resolvable if its blocks can be partitioned into
parallel classes, such that each parallel class partitions the point set of
. Blocks in the same parallel class are parallel. Clearly each parallel class has
blocks.
is affine resolvable, or simply affine, if it can be resolved so that any two nonparallel blocks meet in
points, where
is constant. Affine 1-designs are also called nets. The dual design of a design
is denoted by
. If
and
are both affine, we call
a symmetric net. We use the terminology of Jungnickel [1] (see also [2-5]). In this case
and
. That is,
is an affine
design whose dual
is also affine with the same parameters. For short we call such a symmetric net a
-net.
If
is a symmetric net we shall refer to the parallel classes of
as block classes of
and to the parallel classes of
as point classes of
.
For any finite structure
with point set
and block set
, the code
of
over prime field
is the subspace of the space
of all functions from
to
that is spanned by the incidence vectors of the blocks of
. This code is equivalent to the code given by the column space of any incidence matrix of the incidence structure, where we use the blocks to index the columns (and the points the rows) of the incidence matrix.
2. The Symmetric Net with
and 
The symmetric net that we shall be concerned with in this paper is the one with
and
As a design it has parameters

Its incidence matrix is (1).
A computer search has shown that to within isomorphism there is only one symmetric net with these parameters. We denote this symmetric net by
.
Butson [6] showed that there exist symmetric nets with
any prime and
This was extended to
any prime power by Jungnickel [7]. Therefore
is one of the family of symmetric nets constructed by Jungnickel.
3. The Codes
The columns of the incidence matrix of
can be considered as vectors of the 32-dimensional vector space over any finite prime field
The subspace they generate is the code of the net
over
By computer we found that the binary code (that is, the code over the field of order 2) of
has rank 13. The weight distribution of its codewords is given below. The all one vector is in the code since it is obtained as the sum of the 4 columns corresponding to the blocks of any parallel class in the incidence matrix. Therefore the code is self-complementary in that the complement of a codeword is also a codeword, see [8] or [9]. Hence we only list the number of codewords of weight up to 16.

Since the minimum distance is 8, the binary code is 3-error correcting.
There doesn’t seem to be an easy proof that the dimension of the code is 13 over the binary field. The dimension of the code of
for odd characteristic is 25. This we prove in this paper.
The incidence matrix of
may be put in the form:

where


is an elementary abelian group of order 4.
First suppose that the characteristic of the field is not 2.
The matrices in
can be simultaneously diagonalised by

Conjugating by diagonal
and then by
, the permutation matrix which moves rows (and columns)
to the first eight positions, rows (and columns)
to the next eight positions, rows (and columns)
to the next eight positions, rows (and columns)
to the last eight positions, we get
conjugate to

have determinant 4096. In fact they are Hadamard matrices. Hence the rank of
is
, if the characteristic is not 2.
4. Acknowledgements
The author would like to thank the Deanship of Scientific Research at King Abdulaziz University for supporting a project no. 169/428, where this paper is a part of that project.