4,507 research outputs found
Difference Balanced Functions and Their Generalized Difference Sets
Difference balanced functions from to are closely related
to combinatorial designs and naturally define -ary sequences with the ideal
two-level autocorrelation. In the literature, all existing such functions are
associated with the -homogeneous property, and it was conjectured by Gong
and Song that difference balanced functions must be -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 -homogeneous difference balanced functions. In
particular, we reveal an unexpected equivalence between the -homogeneous
property and multipliers of generalized difference sets. By determining these
multipliers, we prove the Gong-Song conjecture for 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
A classical method of constructing a linear code over \gf(q) with a
-design is to use the incidence matrix of the -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 -designs is studied, and
linear codes with a few weights are obtained from almost difference sets,
difference sets, and a type of -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
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
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
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