编码与密码系列报告十三则

Locally repairable codes (LRCs) are introduced with the aim of reducing the cost of repairing a failed node. An locally repairable code with locality r (r-LRC for short) is a linear code such that every code symbol can be recovered by accessing at most r other code symbols. An r-LRC is called optimal if it achieves the Singleton-type bound. In this talk, we present some new bounds and optimal constructions of locally repairable codes.

Let $p, m, s$ be positive integers such that $p$ is an odd prime number and $m$ is even, let $R=\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}$ ($u^2=0$) where $\mathbb{F}_{p^m}$ is the finite field of $p^m$ elements. Then $R$ is a finite chain ring of $p^{2m}$ elements. Up to now, although the number of all Hermitian self-dual cyclic codes of length $p^s$ over $R$ has been obtained, the effective construction method and explicit expression for all these Hermitian self-dual cyclic codes have not been given in existing literatures.  In this talk, we give an efficient construction for all these Hermitian self-dual cyclic codes by use of column vectors of Kronecker products of matrices with specific types. In particular, we obtain an explicit expression for all distinct Hermitian self-dual cyclic codes of length $p^s$ over $R$, using combination numbers.

We introduce the notion of $\lambda$-constacyclic codes over finite rings $R$ for arbitrary $\lambda$ of $R$. We study the non-invertible-element constacyclic codes (NIE-constacyclic codes) over finite principal ideal rings (PIRs). We first characterize algebraic structures of all NIE-constacyclic codes over finite chain rings and their minimum Hamming distances. A general form of the duals of NIE-constacyclic codes over finite chain rings is also provided. In particular, we provide a necessary and sufficient condition for the dual of an NIE-constacyclic code to be an NIE-constacyclic code. Using the Chinese Remainder Theorem, we obtain algebraic structures and minimum Hamming distances of NIE-constacyclic codes over finite PIRs. Specially, we construct some optimal NIE-constacyclic codes over finite PIRs in the sense that they achieve the maximum possible minimum Hamming distance for some given lengths and cardinalities. This talk is based on joint work with Jingge Liu.

Data hiding or steganography in image is a scheme of hiding the secret message into the cover image. It can be used to transmit the secret data by the cover image or embed important information such as copyright information, authentication information or management information in the cover images. The quality of stego-image is the main object of the applications in copyright protection and the image authentication. The method to improve the quality is to enhance the embedding efficiency. In order to achieve a higher embedding capacity and embedding efficiency, matrix embedding have been proposed using linear block code. This talk introduces the basic theory of the matrix embedding method, and then gives a fast matrix embedding steganography based on the [23,12,7] Golay code, which has high embedding efficiency and considerable payload. The experiment and the analysis results show that this scheme has excellent security performance.

In this talk, we will discuss construction of ternary quantum codes with small distances or small lengths. We also give some hybrid quantum codes with good parameters.

In this talk, we present several new classes of q-ary quantum MDS codes utilizing generalized Reed–Solomon codes satisfying Hermitian self-orthogonal property. Among our constructions, the minimum distance of some q-ary quantum MDS codes can be bigger than q/2 + 1. Comparing to previous known constructions, the lengths of codes in our constructions are more flexible.

Let $F_q$ be the finite field of order $q$. Recently, three distinct eigenvalues of the unitary Cayley graph $\mathcal{C}_{M_n(F_q)}$ have been determined in [Finite Fields Appl. 65 (2020) 101689]. In this talk, completely explicit closed formulas for all the eigenvalues of $\mathcal{C}_{M_n(F_q)}$ are obtained by using a new approach. As applications, the energy, the kirchhoff index and the number of spanning trees of $\mathcal{C}_{M_n(F_q)}$ are derived, respectively. This is a joint work with Jing Huang.

The objective of this talk is to introduce a novel class of sequence pairs, called quasi-orthogonal Z-complementary pairs (QOZCPs), each depicting Z-complementary property for their aperiodic auto-correlation sums and also having a low correlation zone when their aperiodic cross-correlation is considered. Construction of QOZCPs based on Successively Distributed Algorithms under Majorization Minimization (SDAMM) is then proposed. It turns out that QOZCP waveforms are much more Doppler resilient than the known Golay complementary waveforms, and are thus more suitable for use in fully polarimetric radar systems.

In the past few decades, quantum algorithms have become a popular research area of both mathematicians and engineers. Among them, uniform mixing provides a uniform probability distribution of quantum information over time which attracts a special attention. However, there are only a few known examples of graphs which admit uniform mixing. In this talk, a characterization of abelian Cayley graphs having uniform mixing is presented. Some concrete constructions of such graphs are provided. Specifically, for cubelike graphs, it is shown that the Cayley graph ${\rm Cay}(\mathbb{F}_2^{2k};S)$ has uniform mixing if the characteristic function of $S$ is bent. Moreover, a difference-balanced property of the eigenvalues of an abelian Cayley graph having uniform mixing is established. Furthermore, it is proved that an integral abelian Cayley graph exhibits uniform mixing if and only if the underlying group is one of the groups: $\mathbb{Z}_2^d, \mathbb{Z}_3^d$, $\mathbb{Z}_4^d$ or $\mathbb{Z}_2^{r}\otimes \mathbb{Z}_4^d$ for some integers $r \geq 1, d\geq 1$. Thus the classification of integral abelian Cayley graphs having uniform mixing is completed.

APN函数是一类重要的非线性密码函数，是特征2的有限域上抵抗差分密码攻击能力最强的函数。本报告介绍了密码函数的三类安全性指标、密码函数的EA等价与CCZ等价、特征2有限域上APN函数的构造、特征2有限域上APN函数的等价性，在此基础上，我们证明了目前已有的10类多项式APN函数跟APN幂函数是CCZ不等价的。