arXiv subjects

Thor Martinsen

Publications and source records attributed to Thor Martinsen.

5 recordsLinked to original sources

arXiv2016

Bisecting binomial coefficients

In this paper, we deal with the problem of bisecting binomial coefficients. We find many (previously unknown) infinite classes of integers which admit nontrivial bisections, and a class with only trivial bisections. As a byproduct of this last construction, we show conjectures Q2 and Q4 of Cusick and Li. We next find several bounds for the number of nontrivial bisections and further compute (using a supercomputer) the exact number of such bisections for n <= 51.

math.CO↗

arXiv2016

Decomposing generalized bent and hyperbent functions

In this paper we introduce generalized hyperbent functions from $F_{2^n}$ to $Z_{2^k}$, and investigate decompositions of generalized (hyper)bent functions. We show that generalized (hyper)bent functions from $F_{2^n}$ to $Z_{2^k}$ consist of components which are generalized (hyper)bent functions from $F_{2^n}$ to $Z_{2^{k^\prime}}$ for some $k^\prime < k$. For odd $n$, we show that the Boolean functions associated to a generalized bent function form an affine space of semibent functions. This complements a recent result for even $n$, where the associated Boolean functions are bent.

cs.IT↗

arXiv2015

Partial Spread and Vectorial Generalized Bent Functions

In this paper we generalize the partial spread class and completely describe it for generalized Boolean functions from $\F_2^n$ to $\mathbb{Z}_{2^t}$. Explicitly, we describe gbent functions from $\F_2^n$ to $\mathbb{Z}_{2^t}$, which can be seen as a gbent version of Dillon's $PS_{ap}$ class. For the first time, we also introduce the concept of a vectorial gbent function from $\F_2^n$ to $\Z_q^m$, and determine the maximal value which $m$ can attain for the case $q=2^t$. Finally we point to a relation between vectorial gbent functions and relative difference sets.

cs.IT↗

arXiv2015

Generalized bent functions and their Gray images

In this paper we prove that generalized bent (gbent) functions defined on $\mathbb{Z}_2^n$ with values in $\mathbb{Z}_{2^k}$ are regular, and find connections between the (generalized) Walsh spectrum of these functions and their components. We comprehensively characterize generalized bent and semibent functions with values in $\mathbb{Z}_{16}$, which extends earlier results on gbent functions with values in $\mathbb{Z}_4$ and $\mathbb{Z}_8$. We also show that the Gray images of gbent functions with values in $\mathbb{Z}_{2^k}$ are semibent/plateaued when $k=3,4$.

cs.IT↗

arXiv2011

Octal Bent Generalized Boolean Functions

In this paper we characterize (octal) bent generalized Boolean functions defined on $\BBZ_2^n$ with values in $\BBZ_8$. Moreover, we propose several constructions of such generalized bent functions for both $n$ even and $n$ odd.

math.CO↗