Girth-8 regular bipartite graphs can be built with only O(sqrt(n)) check nodes, asymptotically optimal, using sequences free of three-term arithmetic progressions.
The capacity of low-density parity- check codes under message-passing decoding,
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Bounds and New Constructions for Girth-Constrained Regular Bipartite Graphs
Girth-8 regular bipartite graphs can be built with only O(sqrt(n)) check nodes, asymptotically optimal, using sequences free of three-term arithmetic progressions.