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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 ensemble of
such codes based on -partite, -uniform hypergraphs, where depends
only on the code rate, there exist codes satisfying the Varshamov-Gilbert (VG)
and the Costello lower bound on the minimum distance and the free distance,
respectively. A connection between regular bipartite graphs and tailbiting
codes is shown. Some examples of woven graph codes are presented. Among them an
example of a rate woven graph code with
based on Heawood's bipartite graph and containing constituent rate
convolutional codes with overall constraint lengths is
given. An encoding procedure for woven graph codes with complexity proportional
to the number of constituent codes and their overall constraint length
is presented.Comment: Submitted to IEEE Trans. Inform. Theor
Uniform hypergraphs containing no grids
A hypergraph is called an rĆr grid if it is isomorphic to a pattern of r horizontal and r vertical lines, i.e.,a family of sets {A1, ..., Ar, B1, ..., Br} such that Aiā©Aj=Biā©Bj=Ļ for 1ā¤i<jā¤r and {pipe}Aiā©Bj{pipe}=1 for 1ā¤i, jā¤r. Three sets C1, C2, C3 form a triangle if they pairwise intersect in three distinct singletons, {pipe}C1ā©C2{pipe}={pipe}C2ā©C3{pipe}={pipe}C3ā©C1{pipe}=1, C1ā©C2ā C1ā©C3. A hypergraph is linear, if {pipe}Eā©F{pipe}ā¤1 holds for every pair of edges Eā F.In this paper we construct large linear r-hypergraphs which contain no grids. Moreover, a similar construction gives large linear r-hypergraphs which contain neither grids nor triangles. For rā„. 4 our constructions are almost optimal. These investigations are motivated by coding theory: we get new bounds for optimal superimposed codes and designs. Ā© 2013 Elsevier Ltd
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