4 research outputs found

    Some constructions of superimposed codes in Euclidean spaces

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    AbstractWe describe three new methods for obtaining superimposed codes in Euclidean spaces. With help of them we construct codes with parameters improving upon known constructions. We also prove that the spherical simplex code is not optimal as superimposed code at least for dimensions greater than 9

    Uniform hypergraphs containing no grids

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    A hypergraph is called an rƗr 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 Aiāˆ©Aj=Biāˆ©Bj=Ļ† for 1ā‰¤i<jā‰¤r and {pipe}Aiāˆ©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}C1āˆ©C2{pipe}={pipe}C2āˆ©C3{pipe}={pipe}C3āˆ©C1{pipe}=1, C1āˆ©C2ā‰ C1āˆ©C3. A hypergraph is linear, if {pipe}Eāˆ©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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