Optimal Bipartite Ramanujan Graphs from Balanced Incomplete Block Designs: Their Characterizations and Applications to Expander/LDPC Codes

  • Authors:
  • Tom Høholdt;Heeralal Janwal

  • Affiliations:
  • Institute for Mathematics, Technical University of Denmark, Denmark;Department of Mathematics and Computer Science, University of Puerto Rico (UPR), San Juan, 00931 - 3355

  • Venue:
  • AAECC-18 '09 Proceedings of the 18th International Symposium on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes
  • Year:
  • 2009

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Abstract

We characterize optimal bipartite expander graphs and give necessary and sufficient conditions for optimality. We determine the expansion parameters of the BIBD graphs and show that they yield optimal expander graphs that are also bipartite Ramanujan graphs. In particular, we show that the bipartite graphs derived from finite projective and affine geometries yield optimal Ramanujan graphs. This in turn leads to a theoretical explanation of the good performance of a class of LDPC codes.