467 research outputs found
New nonexistence results on -generalized bent functions
In this paper, we present some new nonexistence results on
-generalized bent functions, which improved recent results. More
precisely, we derive new nonexistence results for general and odd or , and further explicitly prove nonexistence of
-generalized bent functions for all integers odd or . The main tools we utilized are certain exponents of minimal
vanishing sums from applying characters to group ring equations that
characterize -generalized bent functions
Nonexistence of two classes of generalized bent functions
We obtain new nonexistence results of generalized bent functions from
\{Z^n}_q to (called type ) in the case that there exist
cyclotomic integers in with absolute value .
This result generalize the previous two scattered nonexistence results
of Pei \cite{Pei} and of Jiang-Deng
\cite{J-D} to a generalized class. In the last section, we remark that this
method can apply to the GBF from to
Nonexistence of Generalized Bent Functions From to
Several nonexistence results on generalized bent functions presented by using some knowledge on cyclotomic number
fields and their imaginary quadratic subfields
On the Non-existence of certain classes of generalized bent functions
We obtain new non-existence results of generalized bent functions from
\ZZ^n_q to \ZZ_q (called type [n,q]). The first case is a class of types where
q=2p_1^{r_1}p_2^{r_2}. The second case contains two types [1 <= n <= 3, 2 *
31^e]$ and [1 <= n <= 7,2 * 151^e].Comment: 15 pages, git commit 20160118a/e17506
On the -Bentness of Boolean Functions
For each non-constant in the set of -variable Boolean functions, the
{\em -transform} of a Boolean function is related to the Hamming
distances from to the functions obtainable from by nonsingular linear
change of basis. Klapper conjectured that no Boolean function exists with its
-transform coefficients equal to (such function is called
-bent). In our early work, we only gave partial results to confirm this
conjecture for small . Here we prove thoroughly that the conjecture is true
by investigating the nonexistence of the partial difference sets in Abelian
groups with special parameters. We also introduce a new family of functions
called almost -bent functions, which are close to -bentness
On Completely Regular Codes
This work is a survey on completely regular codes. Known properties,
relations with other combinatorial structures and constructions are stated. The
existence problem is also discussed and known results for some particular cases
are established. In particular, we present a few new results on completely
regular codes with covering radius 2 and on extended completely regular codes
Necessary Conditions for the Existence of Group-Invariant Butson Matrices and a New Family of Perfect Arrays
Let be a finite abelian group and let denote the least common
multiple of the orders of all elements of . A matrix is a
-invariant matrix whose entries are complex th roots
of unity such that . In this paper, we study the relation
between and so that a matrix exists. We will only focus on
matrices and matrices, where is an odd
prime. By our results, there are open cases left for the existence of
matrices in which . In the last
section, we show that matrices can be used to construct a
new family of perfect polyphase arrays
On -nearly bent Boolean functions
For each non-constant Boolean function , Klapper introduced the notion of
-transforms of Boolean functions. The {\em -transform} of a Boolean
function is related to the Hamming distances from to the functions
obtainable from by nonsingular linear change of basis.
In this work we discuss the existence of -nearly bent functions, a new
family of Boolean functions characterized by the -transform. Let be a
non-affine Boolean function. We prove that any balanced Boolean functions
(linear or non-linear) are -nearly bent if has weight one, which gives a
positive answer to an open question (whether there exist non-affine -nearly
bent functions) proposed by Klapper. We also prove a necessary condition for
checking when a function isn't -nearly bent
On Walsh Spectrum of Cryptographic Boolean Function
Walsh transformation of a Boolean function ascertains a number of cryptographic properties of the Boolean function viz, non-linearity, bentness, regularity, correlation immunity and many more. The functions, for which the numerical value of Walsh spectrum is fixed, constitute a class of Boolean functions known as bent functions. Bent functions possess maximum possible non-linearity and therefore have a significant role in design of cryptographic systems. A number of generalisations of bent function in different domains have been proposed in the literature. General expression for Walsh transformation of generalised bent function (GBF) is derived. Using this condition, a set of Diophantine equations whose solvability is a necessary condition for the existence of GBF is also derived. Examples to demonstrate how these equations can be utilised to establish non-existence and regularity of GBFs is presented
Polynomial Criterion for Abelian Difference Sets
Difference sets are subsets of a group satisfying certain combinatorial
property with respect to the group operation. They can be characterized using
an equality in the group ring of the corresponding group. In this paper, we
exploit the special structure of the group ring of an abelian group to
establish a one-to one correspondence of the class of difference sets with
specific parameters in that group with the set of all complex solutions of a
specified system of polynomial equations. The correspondence also develops some
tests for a Boolean function to be a bent function.Comment: 17 page
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