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Xianhong Xie

Publications and source records attributed to Xianhong Xie.

8 recordsLinked to original sources
arXiv2026

On quadratic binomial vectorial functions with maximal bent components

Assume $n=2m\geq 2$ and let $F(x)=x^{d_1}+x^{d_2}$ be a binomial vectorial function over $\F_{2^n}$ possessing the maximal number (i.e. $2^n-2^m$) of bent components. Suppose the $2$-adic Hamming weights $\wt_2(d_1)$ and $\wt_2(d_2)$ are both at most $2$, we prove that $F(x)$ is affine equivalent to either $x^{2^m+1}$ or $x^{2^i}(x+x^{2^m})$, provided that \[ \ell(n):=\min_{γ:~\F_2(γ)=\F_{2^n}} \dim_{\F_2}\F_2[σ]γ>m, \] where $σ$ is the Frobenius $(x\mapsto x^2)$ on $\F_{2^n}$, and $\gcd(d_1,d_2,2^m-1)>1$. Under this condition, we also establish two bounds on the nonlinearity and the differential uniformity of $F$ by means of the cardinality of its image set.

cs.IT↗
arXiv2024

Binary $[n,(n\pm1)/2]$ cyclic codes with good minimum distances from sequences

Recently, binary cyclic codes with parameters $[n,(n\pm1)/2,\geq \sqrt{n}]$ have been a hot topic since their minimum distances have a square-root bound. In this paper, we construct four classes of binary cyclic codes $\mathcal{C}_{\mathcal{S},0}$, $\mathcal{C}_{\mathcal{S},1}$ and $\mathcal{C}_{\mathcal{D},0}$, $\mathcal{C}_{\mathcal{D},1}$ by using two families of sequences, and obtain some codes with parameters $[n,(n\pm1)/2,\geq \sqrt{n}]$. For $m\equiv2\pmod4$, the code $\mathcal{C}_{\mathcal{S},0}$ has parameters $[2^m-1,2^{m-1},\geq2^{\frac{m}{2}}+2]$, and the code $\mathcal{C}_{\mathcal{D},0}$ has parameters $[2^m-1,2^{m-1},\geq2^{\frac{m}{2}}+2]$ if $h=1$ and $[2^m-1,2^{m-1},\geq2^{\frac{m}{2}}]$ if $h=2$.

cs.IT↗
arXiv2023

Two types of narrow-sense negacyclic BCH codes

Negacyclic BCH codes are an important subclass of negacyclic codes and are the best linear codes in most cases, but their parameters are hard to determine. In this paper, we mainly study two types of negacyclic BCH codes of length $n=\frac{q^{m}-1}{4},\frac{q^{m}+1}{4}$, and give their dimensions and the lower bound on their minimum distance. Furthermore, we provide the weight distribution of narrow-sense neagcyclic BCH codes of length $n=\frac{q^m-1}{4}$ for some special designed distances.

cs.IT↗
arXiv2023

On vectorial functions with maximal number of bent components

We study vectorial functions with maximal number of bent components in this paper. We first study the Walsh transform and nonlinearity of $F(x)=x^{2^e}h(\Tr_{2^{2m}/2^m}(x))$, where $e\geq0$ and $h(x)$ is a permutation over $\F_{2^m}$. If $h(x)$ is monomial, the nonlinearity of $F(x)$ is shown to be at most $ 2^{2m-1}-2^{\lfloor\frac{3m}{2}\rfloor}$ and some non-plateaued and plateaued functions attaining the upper bound are found. This gives a partial answer to the open problems proposed by Pott et al. and Anbar et al. If $h(x)$ is linear, the exact nonlinearity of $F(x)$ is determined. Secondly, we give a construction of vectorial functions with maximal number of bent components from known ones, thus obtain two new classes from the Niho class and the Maiorana-McFarland class. Our construction gives a partial answer to an open problem proposed by Pott et al., and also contains vectorial functions outside the complete Maiorana-McFarland class. Finally, we show that the vectorial function $F: \F_{2^{2m}}\rightarrow \F_{2^{2m}}$, $x\mapsto x^{2^m+1}+x^{2^i+1}$ has maximal number of bent components if and only if $i=0$.

cs.IT↗
arXiv2021

Two Problems about Monomial Bent Functions

In 2008, Langevin and Leander determined the dual function of three classes of monomial bent functions with the help of Stickelberger's theorem: Dillon, Gold and Kasami. In their paper, they proposed one very strong condition such that their method works, and showed that both Gold exponent and Kasami exponent satisfy this condition. In 2018, Pott {\em et al.} investigated the issue of vectorial functions with maximal number of bent components. They found one class of binomial functions which attains the upper bound. They also proposed an open problem regarding monomial function with maximal number of bent components. In this paper, we obtain an interesting result about the condition of Langevin and Leander, and solve the open problem of Pott {\em et al.}. Specifically, we show that: 1) for a monomial bent function over $\mathbb{F}_{2^{2k}}$, if the exponent satisfies the first part of the condition of Langevin and Leander, then it satisfies the entire condition; 2) $x^{2^k+1}$ is the only monomial function over $\mathbb{F}_{2^{2k}}$ which has maximal number of bent components. Fortunately, as a consequence, we also solve an open problem of Ness and Helleseth in 2006.

cs.IT↗
arXiv2016

A Class of Two-Weight and Three-Weight Linear Codes and Their Duals

The objective of this paper is to construct a class of linear codes with two nonzero weights and three nonzero weights by using the general trace functions, which weight distributions has been determined. These linear codes contain some optimal codes, which meets certain bound on linear codes. The dual codes are also studied and proved to be optimal or almost optimal. These codes may have applications in authentication codes, secret sharing schemes and strongly regular graphs.

cs.IT↗