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Constructions of woven graph codes based on constituent block and convolutional codes are studied. It is shown that within the random ensembles of such codes based on s-partite, s-uniform hypergraphs, where s depends only on the code rate, there exist codes satisfying the Gilbert-Varshamov (GV) and the Costello lower bound on the minimum distance and the free distance, respectively. A connection between regular bipartite graphs and tailbiting (TB) codes is shown. Some examples of woven graph codes are presented. Among them, an example of a rate Rwg=1/3 woven graph code with dfree=32 based on Heawood's bipartite graph, containing n=7 constituent rate Rc=2/3 convolutional codes with overall constraint lengths Â¿c =5, is given.