In this paper, we investigated the code over GF(2) which is generated by the incidence matrix of the symmetric (2,4) - net D. By computer search, we found that this binary code of D has rank 13 and the minimum distance is 8.
Symmetric Nets; Codes1. 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 <img src="1-1200035\f4356a4d-42c0-4156-ad0a-2eeef92af252.jpg" /> and <img src="1-1200035\5251e9c6-4a07-4303-a44e-2fb7097769d9.jpg" />
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.
REFERENCESReferencesT. Beth, D. Jungnickel and H. Lenz, “Design Theory,” Cambridge University Press, Cambridge, 1999. C. J. Colbourn and J. H. Dinitz, “The CRC Handbook of Combinatorial Designs,” CRC Press, Boca Raton, New York, London, Tokyo, 1996. Y. J. Ionin and M. S. Shrikhande, “Combinatorics of Symmetric Designs,” Cambridge University Press, Cambridge, 2006. A. N. Al-Kenani and V. C. Mavron, “Non-Tactical Symmetric Nets,” Journal of the London Mathematical Society, Vol. 67, No. 2, 2003, pp. 273-288.
doi:10.1112/S0024610702004052V. C. Mavron and V. D. Tonchev, “On Symmetric Nets and Generalised Hadamard Matrices from Affine Designs,” Journal of Geometry, Vol. 67, No. 1-2, 2000, pp. 180-187. doi:10.1007/BF01220309A. T. Butson, “Generalized Hadamard Matrices,” Proceedings of the American Mathematical Society, Vol. 13, 1962, pp. 894-898.
doi:10.1090/S0002-9939-1962-0142557-0D. Jungnickel, “On Difference Matrices, Resolvable Transversal Designs and Generalised Hadamard Matrices,” Mathematische Zeitschrift, Vol. 167, No. 1, 1979, pp. 49-60. doi:10.1007/BF01215243E. F. Assmus Jr. and J. D. Key, “Designs and Their Codes,” Cambridge Tracts in Mathematics, Vol. 103, Cambridge University Press, 1992. V. D. Tonchev, “Quasi-Symmetric Designs, Codes, Quadrics, and Hyperplane Sections,” Geometriae Dedicata, Vol. 48, No. 3, 1993, pp. 295-308.
doi:10.1007/BF01264073
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
.
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 [
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.
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
.
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.
and 
As a design it has parameters
.
any prime and 
This was extended to 
any prime power by Jungnickel [
is one of the family of symmetric nets constructed by Jungnickel.
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 [
for odd characteristic is 25. This we prove in this paper.
may be put in the form:
is an elementary abelian group of order 4.
can be simultaneously diagonalised by
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
is
, if the characteristic is not 2.