A set of Boolean functions is called a bent set if the sum of any two distinct members is a bent function. We show that any bent set yields a homogeneous system of linked symmetric designs with the ...same design parameters as those systems derived from Kerdock sets. Further we observe that there are bent sets of size equal to the square root of the Kerdock set size which consist of Boolean functions with arbitrary degrees.
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We study permutation polynomials of the shape
G
(
X
)
+
γ
Tr
(
H
(
X
)
)
in
F
p
n
X
. Using a link with functions having a linear structure, we introduce an effective method to construct many such ...permutations, as well as
p-to-1 mappings.
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34.
Crooked binomials Bierbrauer, Jürgen; Kyureghyan, Gohar M.
Designs, codes, and cryptography,
03/2008, Volume:
46, Issue:
3
Journal Article
Peer reviewed
A function
f
:
GF
(2
r
) →
GF
(2
r
) is called crooked if the sets {
f
(
x
) +
f
(
x
+
a
)|
x
∈
GF
(2
r
)} is an affine hyperplane for any nonzero
a
∈
GF
(2
r
). We prove that a crooked binomial ...function
f
(
x
) =
x
d
+
ux
e
defined on
GF
(2
r
) satisfies that both exponents
d
,
e
have 2-weights at most 2.
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35.
Topics in finite fields Kyureghyan, Gohar; Mullen, Gary L; Pott, Alexander
2015, 2015, Volume:
632
Conference Proceeding
Brawley and Carlitz introduced diamond products of elements of finite fields and associated composed products of polynomials in 1987. Composed products yield a method to construct irreducible ...polynomials of large composite degrees from irreducible polynomials of lower degrees. We show that the composed product of two irreducible polynomials of degrees \(m\) and \(n\) is again irreducible if and only if \(m\) and \(n\) are coprime and the involved diamond product satisfies a special cancellation property, the so-called conjugate cancellation. This completes the characterization of irreducible composed products, considered in several previous papers. More generally, we give precise criteria when a diamond product satisfies conjugate cancellation. For diamond products defined via bivariate polynomials, we prove simple criteria that characterize when conjugate cancellation holds. We also provide efficient algorithms to check these criteria. We achieve stronger results as well as more efficient algorithms in the case that the polynomials are bilinear. Lastly, we consider possible constructions of normal elements using composed products and the methods we developed.
On inverses of APN exponents Kyureghyan, G. M.; Suder, V.
2012 IEEE International Symposium on Information Theory Proceedings,
2012-July
Conference Proceeding
Open access
In this extended abstract we present results on the inverses modulo 2 n - 1 of the known APN exponents. In particular, we describe explicitly the inverses of the Welch and Dobbertin exponents and ...give the main ideas of their proofs. Further, we observe that the inverse of the Dobbertin exponent defines an APN function on F 2n of algebraic degree n+3/2, which is the first example of such a function.
Motivated by complexity questions in integer programming, this paper aims to contribute to the understanding of combinatorial properties of integer matrices of row rank \(r\) and with bounded ...subdeterminants. In particular, we study the column number question for integer matrices whose every \(r \times r\) minor is non-zero and bounded by a fixed constant \(\Delta\) in absolute value. Approaching the problem in two different ways, one that uses results from coding theory, and the other from the geometry of numbers, we obtain linear and asymptotically sublinear upper bounds on the maximal number of columns of such matrices, respectively. We complement these results by lower bound constructions, matching the linear upper bound for \(r=2\), and a discussion of a computational approach to determine the maximal number of columns for small parameters \(\Delta\) and \(r\).
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