# American Institute of Mathematical Sciences

November  2008, 2(4): 347-372. doi: 10.3934/amc.2008.2.347

## Zig-zag and replacement product graphs and LDPC codes

 1 Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588, United States 2 1251 Waterfront Place, Seagate Technology, Pittsburgh, PA 15222, United States 3 Institut für Mathematik, Universität Zürich, Zürich, CH-8057

Received  August 2007 Revised  August 2008 Published  November 2008

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.
Citation: Christine A. Kelley, Deepak Sridhara, Joachim Rosenthal. Zig-zag and replacement product graphs and LDPC codes. Advances in Mathematics of Communications, 2008, 2 (4) : 347-372. doi: 10.3934/amc.2008.2.347
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