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Bangteng Xu

Publications and source records attributed to Bangteng Xu.

5 recordsLinked to original sources
arXiv2017

Noncommutative reality-based algebras of rank 6

We classify the RBA-bases of $6$-dimensional noncommutative semisimple algebras for which the algebra has a positive degree map. We show that these RBAs are parametrized by seven real numbers, the first four of which are positive and the remaining three arbitrary. Our classification gives formulas for their standard bases and structure constants. Using these we give a list of all noncommutative integral table algebras of rank 6 with order up to 150. Four in the list are primitive, but we show these cannot be realized as adjacency algebras of association schemes. In the last section of the paper we apply our methods to give a precise description of the noncommutative integral table algebras of rank 6 for which the multiplicity of both linear characters is 1.

math.RA↗
arXiv2016

The recognition problem for table algebras and reality-based algebras

Given a finite-dimensional noncommutative semisimple algebra $A$ with involution, we show that $A$ always has an RBA-basis. We look for an RBA-basis that has integral or rational structure constants, and ask if the RBA admits a positive degree map. For RBAs that have a positive degree map, we try to find an RBA-basis with nonnegative structure constants to determine if there is a generalized table algebra structure. We settle these questions for the algebras $\mathbb{C} \oplus M_n(\mathbb{C})$, $n \ge 2$.

math.RA↗
arXiv2016

Nonlinear functions and difference sets on group actions

Let $G$, $H$ be finite groups and let $X$ be a finite $G$-set. $G$-perfect nonlinear functions from $X$ to $H$ have been studied in several papers. They have more interesting properties than perfect nonlinear functions from $G$ itself to $H$. By introducing the concept of a $(G, H)$-related difference family of $X$, we obtain a characterization of $G$-perfect nonlinear functions on $X$. When $G$ is abelian, we characterize a $G$-difference set of $X$ by the Fourier transform on a normalized $G$-dual set $\widehat X$. We will also investigate the existence and constructions of $G$-perfect nonlinear functions and $G$-bent functions. Several known results in [2,6,10,17] are direct consequences of our results.

math.CO↗
arXiv2014

Fourier Transforms and Bent Functions on Finite Abelian Group-Acted Sets

Let $G$ be a finite abelian group acting faithfully on a finite set $X$. As a natural generalization of the perfect nonlinearity of Boolean functions, the $G$-bentness and $G$-perfect nonlinearity of functions on $X$ are studied by Poinsot et al. [6,7] via Fourier transforms of functions on $G$. In this paper we introduce the so-called $G$-dual set $\widehat X$ of $X$, which plays the role similar to the dual group $\widehat G$ of $G$, and the Fourier transforms of functions on $X$, a generalization of the Fourier transforms of functions on finite abelian groups. Then we characterize the bent functions on $X$ in terms of their own Fourier transforms on $\widehat X$. Bent (perfect nonlinear) functions on finite abelian groups and $G$-bent ($G$-perfect nonlinear) functions on $X$ are treated in a uniform way in this paper, and many known results in [4,2,6,7] are obtained as direct consequences. Furthermore, we will prove that the bentness of a function on $X$ can be determined by its distance from the set of $G$-linear functions. In order to explain the main results clearly, examples are also presented.

cs.DM↗