3,001 research outputs found

    Coding Theory and Algebraic Combinatorics

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    This chapter introduces and elaborates on the fruitful interplay of coding theory and algebraic combinatorics, with most of the focus on the interaction of codes with combinatorial designs, finite geometries, simple groups, sphere packings, kissing numbers, lattices, and association schemes. In particular, special interest is devoted to the relationship between codes and combinatorial designs. We describe and recapitulate important results in the development of the state of the art. In addition, we give illustrative examples and constructions, and highlight recent advances. Finally, we provide a collection of significant open problems and challenges concerning future research.Comment: 33 pages; handbook chapter, to appear in: "Selected Topics in Information and Coding Theory", ed. by I. Woungang et al., World Scientific, Singapore, 201

    Difference Balanced Functions and Their Generalized Difference Sets

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    Difference balanced functions from Fqn∗F_{q^n}^* to FqF_q are closely related to combinatorial designs and naturally define pp-ary sequences with the ideal two-level autocorrelation. In the literature, all existing such functions are associated with the dd-homogeneous property, and it was conjectured by Gong and Song that difference balanced functions must be dd-homogeneous. First we characterize difference balanced functions by generalized difference sets with respect to two exceptional subgroups. We then derive several necessary and sufficient conditions for dd-homogeneous difference balanced functions. In particular, we reveal an unexpected equivalence between the dd-homogeneous property and multipliers of generalized difference sets. By determining these multipliers, we prove the Gong-Song conjecture for qq prime. Furthermore, we show that every difference balanced function must be balanced or an affine shift of a balanced function.Comment: 17 page

    Bounds for DNA codes with constant GC-content

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    We derive theoretical upper and lower bounds on the maximum size of DNA codes of length n with constant GC-content w and minimum Hamming distance d, both with and without the additional constraint that the minimum Hamming distance between any codeword and the reverse-complement of any codeword be at least d. We also explicitly construct codes that are larger than the best previously-published codes for many choices of the parameters n, d and w.Comment: 13 pages, no figures; a few references added and typos correcte

    On a family of strongly regular graphs with λ=1

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