The Code of the Symmetric Net with m = 4 and μ = 2

In this paper, we investigated the code over   2 GF which is generated by the incidence matrix of the symmetric (2,4) net . By computer search, we found that this binary code of has rank 13 and the minimum distance is 8.  


Introduction
is an incidence structure with points, points on each block and any subset of points is contained in exactly  blocks, where > , > 0 . the number of blocks is and the number of blocks on a point is .

b r
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  r  = m v k 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][3][4][5]).In this case  design whose dual is also affine with the same parameters.For short we call such a symmetric net a -net.
is the subspace of the space p F  of all functions from to  p F 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.


The symmetric net that we shall be concerned with in this paper is the one with and = 4 m = 2.

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 2. m =


This was extended to m any prime power by Jungnickel [7].Therefo  is one of the family of symmetric nets constructed by Jung-re -d nickel.

The Codes
The columns of the incidence matrix of  can be considered as vectors of the 32 imensional vector space over any finite prime field .
F The su pace they generate is the code of the net  over .bs F By computer we found that the bina 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 codeord is also a codeword, see [8] or [9].Hence w ry e only e number of codewords of weight up to 16.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 pape The incidence matrix of  may be put in the form: =

I I I I I I I I I I A A B B C C I A B C I A B C I A C B B C A I M I B I B C A C A I B A C A C I B I C B A C I A B I C C I A B B A
, and = 0 1 0 0 1 0 0 0 is an elementary abelian group of or First suppose that the characteristic of the field is not 2.
The matrices in can be simultaneously diagona-  where "−" denotes to −1.
The diagonal block matrices have determinant 4096.In fact they are Hadamard matrices.Hence the rank of M is , if the characteristic is not 2.