Abstract
It is known that the expansion property of a graph influences the performance of the corresponding code when decoded using iterative algorithms. Certain graph products may be used to obtain larger expander graphs from smaller ones. In particular, the zig-zag product and replacement product may be used to construct infinite families of constant degree expander graphs. This paper investigates the use of zig-zag and replacement product graphs for the construction of codes on graphs. A modification of the zig-zag product is also introduced, which can operate on two unbalanced biregular bipartite graphs, and a proof of the expansion property of this modified zig-zag product is presented.
Highlights
Expander graphs are of fundamental interest in mathematics and engineering and have several applications in computer science, complexity theory, designing communication networks, and coding theory [8, 1, 17]
The zig-zag product of two regular graphs is a new graph whose degree is equal to the square of the degree of the second graph and whose expansion property depends on the expansion properties of the two component graphs
In this paper we generalized the zig-zag product resulting in a product suitable for unbalanced bipartite graphs
Summary
Expander graphs are of fundamental interest in mathematics and engineering and have several applications in computer science, complexity theory, designing communication networks, and coding theory [8, 1, 17]. The iterative construction is based on the noncommutative zig-zag graph product introduced by the authors in the same paper. The most prominent example of expander graphs are the class of Ramanujan graphs which are characterized by the property that the second eigenvalue of the adjacency matrix is minimal inside the class of k-regular graphs on n vertices This family of ‘maximal expander graphs’ was independently constructed by Lubotzky, Phillips and Sarnak [10] and by Margulis [11]. In this paper we examine the expansion properties of the zig-zag product and the replacement product in relation to the design of LDPC codes.
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