4,507 research outputs found

    Difference Balanced Functions and Their Generalized Difference Sets

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    Difference balanced functions from FqnF_{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

    Linear Codes from Some 2-Designs

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    A classical method of constructing a linear code over \gf(q) with a tt-design is to use the incidence matrix of the tt-design as a generator matrix over \gf(q) of the code. This approach has been extensively investigated in the literature. In this paper, a different method of constructing linear codes using specific classes of 22-designs is studied, and linear codes with a few weights are obtained from almost difference sets, difference sets, and a type of 22-designs associated to semibent functions. Two families of the codes obtained in this paper are optimal. The linear codes presented in this paper have applications in secret sharing and authentication schemes, in addition to their applications in consumer electronics, communication and data storage systems. A coding-theory approach to the characterisation of highly nonlinear Boolean functions is presented

    Two-tuple balance of non-binary sequences with ideal two-level autocorrelation

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    AbstractLet p be a prime, q=pm and Fq be the finite field with q elements. In this paper, we will consider q-ary sequences of period qn-1 for q>2 and study their various balance properties: symbol-balance, difference-balance, and two-tuple-balance properties. The array structure of the sequences is introduced, and various implications between these balance properties and the array structure are proved. Specifically, we prove that if a q-ary sequence of period qn-1 is difference-balanced and has the “cyclic” array structure then it is two-tuple-balanced. We conjecture that a difference-balanced q-ary sequence of period qn-1 must have the cyclic array structure. The conjecture is confirmed with respect to all of the known q-ary sequences which are difference-balanced, in particular, which have the ideal two-level autocorrelation function when q=p

    Quantum gravity: unification of principles and interactions, and promises of spectral geometry

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    Quantum gravity was born as that branch of modern theoretical physics that tries to unify its guiding principles, i.e., quantum mechanics and general relativity. Nowadays it is providing new insight into the unification of all fundamental interactions, while giving rise to new developments in modern mathematics. It is however unclear whether it will ever become a falsifiable physical theory, since it deals with Planck-scale physics. Reviewing a wide range of spectral geometry from index theory to spectral triples, we hope to dismiss the general opinion that the mere mathematical complexity of the unification programme will obstruct that programme.Comment: This is a contribution to the Proceedings of the 2007 Midwest Geometry Conference in honor of Thomas P. Branson, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

    Kodierungstheorie

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