Edge Transitive Ramanujan Graphs and Highly Symmetric LDPC Good Codes

Algorithms for generation of Ramanujan graphs, other Expanders and related LDPC codes

Expander graphs are highly connected sparse nite graphs. The property of being an expander seems signi cant in many of these mathematical, computational and physical contexts. For practical applications it is very important to construct expander and Ramanujan graphs with given regularity and order. In general, constructions of the best expander graphs with a given regularity and order is no eas...

Product of normal edge-transitive Cayley graphs

For two normal edge-transitive Cayley graphs on groups H and K which have no common direct factor and $gcd(|H/H^prime|,|Z(K)|)=1=gcd(|K/K^prime|,|Z(H)|)$, we consider four standard products of them and it is proved that only tensor product of factors can be normal edge-transitive.

Recent Developments in Low-Density Parity-Check Codes

In this paper we prove two results related to low-density parity-check (LDPC) codes. The first is to show that the generating function attached to the pseudo-codewords of an LDPC code is a rational function, answering a question raised in [6]. The combinatorial information of its numerator and denominator is also discussed. The second concerns an infinite family of q-regular bipartite graphs wi...

Optimal Cycle Codes Constructed From Ramanujan Graphs

We aim here at showing how some known Ramanujan Cayley graphs yield error-correcting codes that are asymptotically optimal in the class of cycle codes of graphs. The main reason why known constructions of Ramanujan graphs yield good cycle codes is that the number of their cycles of a given length behaves essentially like that of random regular graphs. More precisely we show that for actual cons...

Constructions of LDPC Codes using Ramanujan Graphs and Ideas from Margulis