Matrices for graphs, designs and codes
نویسنده
چکیده
The adjacency matrix of a graph can be interpreted as the incidence matrix of a design, or as the generator matrix of a binary code. Here these relations play a central role. We consider graphs for which the corresponding design is a (symmetric) block design or (group) divisible design. Such graphs are strongly regular (in case of a block design) or very similar to a strongly regular graph (in case of a divisible design). Many construction and properties for these kind of graphs are obtained. We also consider the binary code of a strongly regular graph, work out some theory and give several examples.
منابع مشابه
Some Optimal Codes From Designs
The binary and ternary codes spanned by the rows of the point by block incidence matrices of some 2-designs and their complementary and orthogonal designs are studied. A new method is also introduced to study optimal codes.
متن کاملPermutation decoding for codes from designs, finite geometries and graphs
The method of permutation decoding was first developed by MacWilliams in the early 60’s and can be used when a linear code has a sufficiently large automorphism group to ensure the existence of a set of automorphisms, called a PD-set, that has some specifed properties. These talks will describe some recent developments in finding PD-sets for codes defined through the row-span over finite fields...
متن کاملRegular low-density parity-check codes from oval designs
This paper presents a construction of low-density parity-check (LDPC) codes based on the incidence matrices of oval designs. The new LDPC codes have regular parity-check matrices and Tanner graphs free of 4-cycles. Like the finite geometry codes, the codes from oval designs have parity-check matrices with a large proportion of linearly dependent rows and can achieve significantly better minimum...
متن کاملHigh-rate LDPC codes from unital designs
This paper presents a construction of very high-rate low-density parity-check (LDPC) codes based on the incidence matrices of unital designs. Like the projective geometry and oval designs, unital designs exist with incidence matrices which are significantly rank deficient. Thus high-rate LDPC codes with a large number of linearly dependent parity-check equations can be constructed. The LDPC cod...
متن کاملA Family of Antipodal Distance-Regular Graphs Related to the Classical Preparata Codes
A new family of distance-regular graphs is constructed. They are antipodal 2 -fold covers of the complete graph on 2 vertices. The automorphism groups are determined, and the extended Preparata codes are reconstructed using walks on these graphs. There are connections to other interesting structures: the graphs are equivalent to certain generalized Hadamard matrices; and their underlying 3-clas...
متن کاملA Family of Antipodal Distance-Regular Graphs Related to the Classical Preparata Codes
A new family of distance-regular graphs is constructed. They are antipodal 22t−1-fold covers of the complete graph on 22t vertices. The automorphism groups are determined, and the extended Preparata codes are reconstructed using walks on these graphs. There are connections to other interesting structures: the graphs are equivalent to certain generalized Hadamard matrices; and their underlying 3...
متن کامل