Cycle’s topological optimizations and the iterative decoding problem on general graphs

We survey fundamental problem of codes in graph theory. We report on the state of the art solution of the cycle's topology optimization using progressive column elimination and special labeling approach. This, in particular, allows constructing finite-length Quasi-cyclic low-density parity-check codes which has error correcting capability close to graph threshold and achieves low error floors properties. We formulate several further codes on the graph construction problems and provide analysis of possible ways to solve it.