Almost Combinatorial Selector Sets

Classes of subsets of positive integers based on the notion of selector functions similar to the Jockusch selector of a semirecursive set are studied.     

迹非零布尔矩阵的幂敛指数集

本文给出了恰含d个非零对角元的n阶布尔矩阵类（1≤d≤n)的幂敛指数集的一个明显表达式。     

关于两个P-值逻辑函数的和函数的Chrestenson谱公式

类似于两个布尔函数的和函数的ｗａｌｓｈ谱公式,本文给出了两个３－值、５－值、７－值逻辑函数和函数的Ｃｈｒｅｓｔｅｎｓｏｎ谱公式。     

计算OC型逻辑函数的布尔偏导数与布尔差分的表格方法

讨论了OC型逻辑函数一阶布尔偏导数的规范OC展开式．在此基础上提出了计算OC型逻辑函数一阶、二阶布尔偏导数的表格方法．根据二阶布尔差分的定义以及二阶布尔差分与布尔偏导数的关系提出了两种计算OC型逻辑函数二阶布尔差分的表格方法,并作了比较．本文提出的计算OC型逻辑函数布尔偏导数与布尔差分的表格方法具有简捷、方便以及易于计算机编程操作等优点．     

关于蕴涵BCK—代数的伴随半群

我们证明了蕴涵BCK-代数的伴随半群是一个上半格;具有条件（s)的蕴涵BCK-代数的伴随半群是一个广义布尔代数。更进一步证明了有界蕴涵BCK-代数的伴随半群是一个布尔代数。     

The covering radius of the Reed–Muller code RM(2,7) is 40

It was proved by J. Schatz that the covering radius of the second order Reed–Muller code $RM(2,6)$ is 18 (Schatz (1981)). However, the covering radius of $RM(2,7)$ has been an open problem for many years. In this paper, we prove that the covering radius of $RM(2,7)$ is 40, which is the same as the covering radius of $RM(2,7)$ in $RM(3,7)$. As a corollary, we also find new upper bounds for the covering radius of $RM(2,n)$, $n=8,9,10$.     

The crooked property

Crooked permutations were introduced twenty years ago to construct interesting objects in graph theory. These functions, over ${\mathbb{F}}_{2^n}$ with odd n, are such that their derivatives have as image set a complement of a hyperplane. The field of applications was extended later, in particular to cryptography. However binary crooked functions are rare. It is still unknown if non quadratic crooked functions do exist. We extend the concept and propose to study the crooked property for any characteristic. A function F, from ${\mathbb{F}}_{p^n}$ to itself, satisfies this property if all its derivatives have as image set an affine subspace. We show that the partially-bent vectorial functions and the functions satisfying the crooked property are strongly related. We later focus on the components of these functions, establishing that the existence of linear structures is here decisive. We then propose a symbolic approach to identify the linear structures. We claim that this problem consists in solving a system of linear equations, and can often be seen as a combinatorial problem.     

Hyperarithmetical Boolean Algebras with a Distinguished Ideal

We prove a general theorem that allows us to pass from a hyperarithmetical Boolean algebra with a distinguished ideal to some computable Boolean algebra connected with the former by natural algebraic operations. Some examples are given.     

一类本原无限布尔方阵的本原指数集的刻划

引入了本原无限布尔方阵的概念,给出了无限布尔方阵为本原阵的一个充分必要条件,最后给出了一类本原无限布尔方阵的本原指数集的刻划.