Bent Rectangles

Classifying bent functions by their Cayley graphs

In 1999 Bernasconi and Codenotti noted that the Cayley graph of a bent function is strongly regular. This paper describes the concept of extended Cayley equivalence of bent functions, discusses some connections between bent functions, designs, and codes, and explores the relationship between extended Cayley equivalence and extended affine equivalence. SageMath scripts and CoCalc worksheets are used to compute and display some of these relationships, for bent functions up to dimension 8.

Affine type A crystal structure on tensor products of rectangles, Demazure characters, and nilpotent varieties

Answering a question of Kuniba, Misra, Okado, Takagi, and Uchiyama, it is shown that certain Demazure characters of affine type A, coincide with the graded characters of coordinate rings of closures of conjugacy classes of nilpotent matrices. This entails a translation of the affine type A crystal theory into the language of tableaux following Nakayashiki and Yamada, for the case of tensor products of the classical crystals indexed by rectangular partitions. In particular the explicit action of the zero-th crystal raising operator on the above crystals is given, and its direct connection with …

On $q$-nearly bent Boolean functions

For each non-constant Boolean function $q$, Klapper introduced the notion of $q$-transforms of Boolean functions. The {\em $q$-transform} of a Boolean function $f$ is related to the Hamming distances from $f$ to the functions obtainable from $q$ by nonsingular linear change of basis. In this work we discuss the existence of $q$-nearly bent functions, a new family of Boolean functions characterized by the $q$-transform. Let $q$ be a non-affine Boolean function. We prove that any balanced Boolean functions (linear or non-linear) are $q$-nearly bent if $q$ has weight one, which gives a positive a…

On constructions and properties of self-dual generalized bent functions

Bent functions of the form $\mathbb{F}_2^n\rightarrow\mathbb{Z}_q$, where $q\geqslant2$ is a positive integer, are known as generalized bent (gbent) functions. Gbent functions for which it is possible to define a dual gbent function are called regular. A regular gbent function is said to be self-dual if it coincides with its dual. In this paper we explore self-dual gbent functions for even $q$. We consider several primary and secondary constructions of such functions. It is proved that the numbers of self-dual and anti-self dual gbent functions coincide. We give necessary and sufficient condit…

The down operator and expansions of near rectangular k-Schur functions

We prove that the Lam-Shimozono "down operator" on the affine Weyl group induces a derivation of the affine Fomin-Stanley subalgebra of the affine nilCoxeter algebra. We use this to verify a conjecture of Berg, Bergeron, Pon and Zabrocki describing the expansion of k-Schur functions of "near rectangles" in the affine nilCoxeter algebra. Consequently, we obtain a combinatorial interpretation of the corresponding k-Littlewood--Richardson coefficients.

关于 bent 函数数量的一个渐近下界

An asymptotic lower bound on the number of bent functions

本文通过改进 Maiorana–McFarland bent 函数族并结合 latin squares 与 hypercubes 中 transversal 计数进展，建立了 Boolean bent 函数数量的新渐近下界。

Cubic bent functions outside the completed Maiorana-McFarland class

In this paper we prove that in opposite to the cases of 6 and 8 variables, the Maiorana-McFarland construction does not describe the whole class of cubic bent functions in $n$ variables for all $n\ge 10$. Moreover, we show that for almost all values of $n$, these functions can simultaneously be homogeneous and have no affine derivatives.

Several new classes of Boolean functions with few Walsh transform values

In this paper, several new classes of Boolean functions with few Walsh transform values, including bent, semi-bent and five-valued functions, are obtained by adding the product of two or three linear functions to some known bent functions.Numerical results show that the proposed class contains cubic bent functions that are affinely inequivalent to all known quadratic ones. Meanwhile, we determine the distribution of the Walsh spectrum of five-valued functions constructed in this paper.

几乎Maiorana-McFarland bent函数

Almost Maiorana-McFarland bent functions

本文研究了广义Maiorana-McFarland类中的几乎Maiorana-McFarland bent函数，刻画其bent性质、确定对偶，提出构造算法生成${M}^\n#$外函数，解决开放问题并实现类内外函数的相互生成。

关于向量布尔函数与仿射函数之间的最小Hamming距离

On the minimum Hamming distance between vectorial Boolean and affine functions

本文研究向量布尔函数与仿射函数的Hamming距离，证明了2017年关于该度量的猜想上界对两类向量bent函数成立，改进了单项式函数的已知界，并计算了维数不超过9的APN函数到仿射函数的精确距离。

关于Maiorana-McFarland类扩展的研究

On the Maiorana-McFarland Class Extensions

本文研究Maiorana-McFarland类及其闭包和扩展的基数、度量性质，给出显式公式、上界及渐近紧界，并计算平均仿射子空间数，揭示扩展与闭包间的差异。

有限阿贝尔群上过约束字符和与广义弯曲、平坦和景观函数的分解

Overconstrained character sums over finite abelian groups and decompositions of generalized bent, plateaued and landscape functions

本文研究了有限阿贝尔群上的过约束字符和，提出了一种新的2^ℓ进制表示方法，并探讨了广义弯曲、平坦和景观函数的分解及其性质。