arXiv subjects

Yongbo Xia

Publications and source records attributed to Yongbo Xia.

8 recordsLinked to original sources

arXiv2026

Strong Singleton-Like Bounds, Quasi-Perfect Codes and Distance-Optimal Codes in the Sum-Rank Metric

Codes in the sum-rank metric have received many attentions in recent years, since they have wide applications in the multishot network coding, the space-time coding and the distributed storage. In this paper, by constructing covering codes in the sum-rank metric from covering codes in the Hamming metric, we derive new upper bounds on sizes, the covering radii and the block length functions of codes in the sum-rank metric. As applications, we present several strong Singleton-like bounds that are tighter than the classical Singleton-like bound when block lengths are large. In addition, we give the explicit constructions of the distance-optimal sum-rank codes of matrix sizes $s\times s$ and $2\times 2$ with minimum sum-rank distance four respectively by using cyclic codes in the Hamming metric. More importantly, we present an infinite families of quasi-perfect $q$-ary sum-rank codes with matrix sizes $2\times m$. Furthermore, we construct almost MSRD codes with larger block lengths and demonstrate how the Plotkin sum can be used to give more distance-optimal sum-rank codes.

cs.IT↗

arXiv2024

Further Investigation on Differential Properties of the Generalized Ness-Helleseth Function

Let $n$ be an odd positive integer, $p$ be a prime with $p\equiv3\pmod4$, $d_{1} = {{p^{n}-1}\over {2}} -1 $ and $d_{2} =p^{n}-2$. The function defined by $f_u(x)=ux^{d_{1}}+x^{d_{2}}$ is called the generalized Ness-Helleseth function over $\mathbb{F}_{p^n}$, where $u\in\mathbb{F}_{p^n}$. It was initially studied by Ness and Helleseth in the ternary case. In this paper, for $p^n \equiv 3 \pmod 4$ and $p^n \ge7$, we provide the necessary and sufficient condition for $f_u(x)$ to be an APN function. In addition, for each $u$ satisfying $χ(u+1) = χ(u-1)$, the differential spectrum of $f_u(x)$ is investigated, and it is expressed in terms of some quadratic character sums of cubic polynomials, where $χ(\cdot)$ denotes the quadratic character of $\mathbb{F}_{p^n}$.

cs.CR↗

arXiv2022

On the Differential Properties of the Power Mapping $x^{p^m+2}$

Let $m$ be a positive integer and $p$ a prime. In this paper, we investigate the differential properties of the power mapping $x^{p^m+2}$ over $\mathbb{F}_{p^n}$, where $n=2m$ or $n=2m-1$. For the case $n=2m$, by transforming the derivative equation of $x^{p^m+2}$ and studying some related equations, we completely determine the differential spectrum of this power mapping. For the case $n=2m-1$, the derivative equation can be transformed to a polynomial of degree $p+3$. The problem is more difficult and we obtain partial results about the differential spectrum of $x^{p^m+2}$.

cs.CR↗

arXiv2021

The Differential Spectrum of the Power Mapping $x^{p^n-3}$

Let $n$ be a positive integer and $p$ a prime. The power mapping $x^{p^n-3}$ over $\mathbb{F}_{p^n}$ has desirable differential properties, and its differential spectra for $p=2,\,3$ have been determined. In this paper, for any odd prime $p$, by investigating certain quadratic character sums and some equations over $\mathbb{F}_{p^n}$, we determine the differential spectrum of $x^{p^n-3}$ with a unified approach. The obtained result shows that for any given odd prime $p$, the differential spectrum can be expressed explicitly in terms of $n$. Compared with previous results, a special elliptic curve over $\mathbb{F}_{p}$ plays an important role in our computation for the general case $p \ge 5$.

cs.IT↗

arXiv2017

A new class of permutation trinomials constructed from Niho exponents

Permutation polynomials over finite fields are an interesting subject due to their important applications in the areas of mathematics and engineering. In this paper we investigate the trinomial $f(x)=x^{(p-1)q+1}+x^{pq}-x^{q+(p-1)}$ over the finite field $\mathbb{F}_{q^2}$, where $p$ is an odd prime and $q=p^k$ with $k$ being a positive integer. It is shown that when $p=3$ or $5$, $f(x)$ is a permutation trinomial of $\mathbb{F}_{q^2}$ if and only if $k$ is even. This property is also true for more general class of polynomials $g(x)=x^{(q+1)l+(p-1)q+1}+x^{(q+1)l+pq}-x^{(q+1)l+q+(p-1)}$, where $l$ is a nonnegative integer and $\gcd(2l+p,q-1)=1$. Moreover, we also show that for $p=5$ the permutation trinomials $f(x)$ proposed here are new in the sense that they are not multiplicative equivalent to previously known ones of similar form.

cs.IT↗

arXiv2017

Another Generalization of the Reed-Muller Codes

The punctured binary Reed-Muller code is cyclic and was generalized into the punctured generalized Reed-Muller code over $\gf(q)$ in the literature. The major objective of this paper is to present another generalization of the punctured binary Reed-Muller code. Another objective is to construct a family of reversible cyclic codes that are related to the newly generalized Reed-Muller codes.

cs.IT↗

arXiv2016

On the Correlation Distribution for a Ternary Niho Decimation

In this paper, let $n=2m$ and $d=3^{m+1}-2$ with $m\geq2$ and $\gcd(d,3^n-1)=1$. By studying the weight distribution of the ternary Zetterberg code and counting the numbers of solutions of some equations over the finite field $\mathbb{F}_{3^n}$, the correlation distribution between a ternary $m$-sequence of period $3^n-1$ and its $d$-decimation sequence is completely determined. This is the first time that the correlation distribution for a non-binary Niho decimation has been determined since 1976.

cs.IT↗

arXiv2016

Three-Weight Ternary Linear Codes from a Family of Monomials

Based on a generic construction, two classes of ternary three-weight linear codes are obtained from a family of power functions, including some APN power functions. The weight distributions of these linear codes are determined through studying the properties of some exponential sum related to the proposed power functions.

cs.IT↗