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Bent functions, SDP designs and their automorphism groups
PhD ThesisIn a 1976 paper Rothaus coined the term “bent” to describe a function f from a vector space V (n, 2) to F2 with the property that the Fourier coefficients of (−1)f have unit magnitude. Such a function has the maximum possible distance from the set of linear functions, hence the name, and has useful correlation properties. These lead to various applications to coding theory and cryptography, some of which are
described. A standard notion of the equivalence of two bent functions is discussed and related to the coding theory setting.
Two constructions mentioned by Rothaus and generalised by Maiorana are described. A further generalisation of one of these, involving sets of bent functions on direct summands of the original vector space, is described and proved. Various methods including computer searches are used to find appropriate sets of bent functions and hence many new equivalence classes of bent functions of 8 variables.
Equivalence class invariants are used to show that most of these classes cannot be constructed by the earlier methods. Some bounds on numbers of bent functions are discussed.
A 2-design is said to have the symmetric difference property (SDP) if the symmetric difference of any three blocks is either a block or the complement of a block — such a design is very close to being a 3-design. All SDP designs are induced by bent functions, and conversely. Work on the automorphism groups of various SDP designs involving computer algebra is described. An SDP design on 256 points with
trivial automorphism group is noted.
Some connections with strongly-regular graphs are discussed. An infinite class of pseudo-geometric strongly-regular graphs induced by bent functions is noted, and bent functions which are their own Fourier transform duals are investigated. Finally, some open problems and ideas for future work are described
Strongly Regular Graphs Constructed from -ary Bent Functions
In this paper, we generalize the construction of strongly regular graphs in
[Y. Tan et al., Strongly regular graphs associated with ternary bent functions,
J. Combin.Theory Ser. A (2010), 117, 668-682] from ternary bent functions to
-ary bent functions, where is an odd prime. We obtain strongly regular
graphs with three types of parameters. Using certain non-quadratic -ary bent
functions, our constructions can give rise to new strongly regular graphs for
small parameters.Comment: to appear in Journal of Algebraic Combinatoric
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
Semifields, relative difference sets, and bent functions
Recently, the interest in semifields has increased due to the discovery of
several new families and progress in the classification problem. Commutative
semifields play an important role since they are equivalent to certain planar
functions (in the case of odd characteristic) and to modified planar functions
in even characteristic. Similarly, commutative semifields are equivalent to
relative difference sets. The goal of this survey is to describe the connection
between these concepts. Moreover, we shall discuss power mappings that are
planar and consider component functions of planar mappings, which may be also
viewed as projections of relative difference sets. It turns out that the
component functions in the even characteristic case are related to negabent
functions as well as to -valued bent functions.Comment: Survey paper for the RICAM workshop "Emerging applications of finite
fields", 09-13 December 2013, Linz, Austria. This article will appear in the
proceedings volume for this workshop, published as part of the "Radon Series
on Computational and Applied Mathematics" by DeGruyte
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