Symmetric configurations for bipartite-graph codes
Alexander Davydov
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Abstract
We propose geometrical methods for constructing square 01-matrices with the same number n of units in every row and column, and such that any two rows of the matrix have at most one unit in the same position. In terms of Design Theory, such a matrix is an incidence matrix of a symmetric configuration. Also, it gives rise to an n-regular bipartite graphs without 4-cycles, which can be used for constructing bipartite-graph codes so that both the classes of their vertices are associated with local constraints (constituent codes). We essentially extend the region of parameters of such matrices by using some results from Galois Geometries. Many new matrices are either circulant or consist of circulant submatrices: this provides code parity-check matrices consisting of circulant submatrices, and hence quasi-cyclic bipartite-graph codes with simple implementation.
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