线性码与bent函数及其推广的关联结构

向量负弯曲概念：相似性、差异性与推广

Vectorial Negabent Concepts: Similarities, Differences, and Generalizations

本文澄清了两种向量负弯曲概念的异同，将其推广至广义布尔函数，并利用置换构造了Z_{2^k}-弯曲函数。

Secondary Constructions of Bent Functions and Highly Nonlinear Resilient Functions

In this paper, we first present a new secondary construction of bent functions (building new bent functions from two already defined ones). Furthermore, we apply the construction using as initial functions some specific bent functions and then provide several concrete constructions of bent functions. The second part of the paper is devoted to the constructions of resilient functions. We give a generalization of the indirect sum construction for constructing resilient functions with high nonlinearity. In addition, we modify the generalized construction to ensure a high nonlinearity of the const…

Bent and $\mathbb Z_{2^k}$-bent functions from spread-like partitions

Bent functions from a vector space $V_n$ over $\mathbb F_2$ of even dimension $n=2m$ into the cyclic group $\mathbb Z_{2^k}$, or equivalently, relative difference sets in $V_n\times\mathbb Z_{2^k}$ with forbidden subgroup $\mathbb Z_{2^k}$, can be obtained from spreads of $V_n$ for any $k\le n/2$. In this article, existence and construction of bent functions from $V_n$ to $\mathbb Z_{2^k}$, which do not come from the spread construction is investigated. A construction of bent functions from $V_n$ into $\mathbb Z_{2^k}$, $k\le n/6$, (and more generally, into any abelian group of order $2^k$) is…

A Construction of Linear Codes over $\f_{2^t}$ from Boolean Functions

In this paper, we present a construction of linear codes over $\f_{2^t}$ from Boolean functions, which is a generalization of Ding's method \cite[Theorem 9]{Ding15}. Based on this construction, we give two classes of linear codes $\tilde{\C}_{f}$ and $\C_f$ (see Theorem \ref{thm-maincode1} and Theorem \ref{thm-maincodenew}) over $\f_{2^t}$ from a Boolean function $f:\f_{q}\rightarrow \f_2$, where $q=2^n$ and $\f_{2^t}$ is some subfield of $\f_{q}$. The complete weight enumerator of $\tilde{\C}_{f}$ can be easily determined from the Walsh spectrum of $f$, while the weight distribution of the co…

从独立结构生成丝带：超收缩性、Φ-互信息和矩阵Φ熵

Ribbons from Independence Structure: Hypercontractivity, $Φ$-Mutual Information, and Matrix $Φ$-Entropy

本文研究了基于独立结构的超收缩性与Φ-互信息，提出矩阵Φ-丝带及其性质，推导了非平凡点并计算了精确常数。