Woven Graph Codes: Asymptotic Performances and Examples

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 $R_{\rm wg}=1/3$ woven graph code with $d_{\rm free}=32$ based on Heawood's bipartite graph, containing $n=7$ constituent rate $R^{\rm c}=2/3$ convolutional codes with overall constraint lengths $\nu^{\rm c}=5$, is given.