Some combinatorial aspects of constructing bipartite-graph codes

TL;DR

This paper introduces geometrical methods to construct bipartite-graph codes using special matrices and graphs, significantly expanding the parameters for such codes and enabling quasi-cyclic implementations for efficient decoding.

Contribution

It presents new geometrical techniques for constructing bipartite-graph codes with expanded parameters and circulant structures for easier implementation.

Findings

Expanded the range of parameters for n-regular bipartite graphs without 4-cycles.

Constructed matrices with circulant submatrices for quasi-cyclic code design.

Enhanced the lengths and rates of bipartite-graph codes.

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 contain at most one unit in common. These matrices are equivalent to n-regular bipartite graphs without 4-cycles, and therefore can be used for the construction of efficient bipartite-graph codes such that both the classes of its vertices are associated with local constraints. We significantly extend the region of parameters m,n for which there exist an n-regular bipartite graph with 2m vertices and without 4-cycles. In that way we essentially increase the region of lengths and rates of the corresponding bipartite-graph codes. 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.