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Yanbin Zheng

Publications and source records attributed to Yanbin Zheng.

12 recordsLinked to original sources
arXiv2026

Non-GRS type MDS and AMDS codes from extended TGRS codes

Maximum distance separable (MDS) and almost maximum distance separable (AMDS) codes have been widely used in various fields such as communication systems, data storage, and quantum codes because of their algebraic properties and excellent error-correcting capabilities. In this paper, we construct a class of extended twisted generalized Reed-Solomon (TGRS) codes and determine the necessary and sufficient conditions for these codes to be MDS or AMDS. Additionally, we prove that these codes are not equivalent to generalized Reed-Solomon (GRS) codes. As an application, under certain circumstances, we compute the covering radii and deep holes of these codes.

cs.IT↗
arXiv2025

Large class of many-to-one mappings over quadratic extension of finite fields

Many-to-one mappings and permutation polynomials over finite fields have important applications in cryptography and coding theory. In this paper, we study the many-to-one property of a large class of polynomials such as $f(x) = h(a x^q + b x + c) + u x^q + v x$, where $h(x) \in \mathbb{F}_{q^2}[x]$ and $a$, $b$, $c$, $u$, $v \in \mathbb{F}_{q^2}$. Using a commutative diagram satisfied by $f(x)$ and trace functions over finite fields, we reduce the problem whether $f(x)$ is a many-to-one mapping on $\mathbb{F}_{q^2}$ to another problem whether an associated polynomial $g(x)$ is a many-to-one mapping on the subfield $\mathbb{F}_{q}$. In particular, when $h(x) = x^{r}$ and $r$ satisfies certain conditions, we reduce $g(x)$ to polynomials of small degree or linearized polynomials. Then by employing the many-to-one properties of these low degree or linearized polynomials on $\mathbb{F}_{q}$, we derive a series of explicit characterization for $f(x)$ to be many-to-one on $\mathbb{F}_{q^2}$. On the other hand, for all $1$-to-$1$ mappings obtained in this paper, we determine the inverses of these permutation polynomials. Moreover, we also explicitly construct involutions from $2$-to-$1$ mappings of this form. Our findings generalize and unify many results in the literature.

cs.IT↗
arXiv2025

On many-to-one property of generalized cyclotomic mappings

The generalized cyclotomic mappings over finite fields $\mathbb{F}_{q}$ are those mappings which induce monomial functions on all cosets of an index $\ell$ subgroup $C_0$ of the multiplicative group $\mathbb{F}_{q}^{*}$. Previous research has focused on the one-to-one property, the functional graphs, and their applications in constructing linear codes and bent functions. In this paper, we devote to study the many-to-one property of these mappings. We completely characterize many-to-one generalized cyclotomic mappings for $1 \le \ell \le 3$. Moreover, we completely classify $2$-to-$1$ generalized cyclotomic mappings for any divisor $\ell$ of $q-1$. In addition, we construct several classes of many-to-one binomials and trinomials of the form $x^r h(x^{q-1})$ on $\mathbb{F}_{q^2}$, where $h(x)^{q-1}$ induces monomial functions on the cosets of a subgroup of $U_{q+1}$.

cs.IT↗
arXiv2024

On many-to-one mappings over finite fields

We introduce the definition of $m$-to-$1$ mappings between two finite sets, which unifies and generalizes the definitions of $2$-to-$1$ and $n$-to-$1$ mappings in recent literature. We also characterize these $m$-to-$1$ mappings in terms of the generalized local criterion and thus provide three generic constructions of $m$-to-$1$ mappings, which unify and generalize the previous known constructions. Using these constructions, the problem whether $x^r h(x^s)$ is $m$-to-$1$ on the multiplicative group $\mathbb{F}_{q}^{*}$ is converted into that whether an associated polynomial $x^{r_1} h(x)^{s_1}$ is $m_2$-to-$1$ on the order~$\ell$ subgroup~$U_{\ell}$ of $\mathbb{F}_{q}^{*}$, where $m_2 = m / (r, s)$ and $\ell = (q-1) / s$. Furthermore, the $m_2$-to-$1$ property of $x^{r_1} h(x)^{s_1}$ on $U_{\ell}$ is studied in detail in four different cases. In addition, a recursive construction of $m$-to-$1$ mappings from $m$-to-$1$ mappings is proposed.

cs.IT↗
arXiv2024

Experimental Investigation of an Incremental Contact Model for Hyperelastic Solids Using In-Situ Optical Interferometric Technique

The hyperelastic materials would contribute to the intricacies of rough surface contact, primarily due to the heightened nonlinearity caused by stress concentration. In our previous research, an incremental contact model tailored for hyperelastic materials is proposed and validated by finite element (FEM) simulations. From an experimental perspective, this study employs an in-situ optical interferometric technique to precisely document the actual contact zone between hyperelastic solids and quartz glass. Simultaneously, the contact force is meticulously recorded in sync by a force sensor positioned beneath the hyperelastic samples. Comparing with the predictions of incremental contact model for hyperelastic materials, a significant agreement becomes evident, almost in a range of nearly complete contact. Its significance extends to practical domains such as sealing mechanisms, leakage prevention, and structural integrity, offering valuable insights for these applications.

cond-mat.soft↗
arXiv2023

Dynamic Efficient Adversarial Training Guided by Gradient Magnitude

Adversarial training is an effective but time-consuming way to train robust deep neural networks that can withstand strong adversarial attacks. As a response to its inefficiency, we propose Dynamic Efficient Adversarial Training (DEAT), which gradually increases the adversarial iteration during training. We demonstrate that the gradient's magnitude correlates with the curvature of the trained model's loss landscape, allowing it to reflect the effect of adversarial training. Therefore, based on the magnitude of the gradient, we propose a general acceleration strategy, M+ acceleration, which enables an automatic and highly effective method of adjusting the training procedure. M+ acceleration is computationally efficient and easy to implement. It is suited for DEAT and compatible with the majority of existing adversarial training techniques. Extensive experiments have been done on CIFAR-10 and ImageNet datasets with various training environments. The results show that the proposed M+ acceleration significantly improves the training efficiency of existing adversarial training methods while achieving similar robustness performance. This demonstrates that the strategy is highly adaptive and offers a valuable solution for automatic adversarial training.

cs.LG↗
arXiv2020

On inverse of permutation polynomials of small degree over finite fields, II

We investigate the permutation property of polynomials of the form $x^{r}(x^{s} -a)^{t}$, and give the expressions of their inverses. In particular, explicit expressions of inverses of permutation polynomials $x(x^3 -a)^2$ and $x(x^2 -a)^3$ on $\mathbb{F}_{7^n}$ are presented. Then, using some known results, we obtain the inverses of all permutation polynomials of degree $6, 7, 8$ over finite fields.

math.NT↗
arXiv2020

On Inverses of Permutation Polynomials of Small Degree over Finite Fields

Permutation polynomials (PPs) and their inverses have applications in cryptography, coding theory and combinatorial design theory. In this paper, we make a brief summary of the inverses of PPs of finite fields, and give the inverses of all PPs of degree $\leq 6$ over finite fields $\mathbb{F}_{q}$ for all $q$ and the inverses of all PPs of degree $7$ over $\mathbb{F}_{2^n}$. The explicit inverse of a class of fifth degree PPs is the main result, which is obtained by using Lucas' theorem, some congruences of binomial coefficients, and a known formula for the inverses of PPs of finite fields.

math.CO↗
arXiv2018

Large classes of permutation polynomials over $\mathbb{F}_{q^2}$

Permutation polynomials (PPs) of the form $(x^{q} -x + c)^{\frac{q^2 -1}{3}+1} +x$ over $\mathbb{F}_{q^2}$ were presented by Li, Helleseth and Tang [Finite Fields Appl. 22 (2013) 16--23]. More recently, we have constructed PPs of the form $(x^{q} +bx + c)^{\frac{q^2 -1}{d}+1} -bx$ over $\mathbb{F}_{q^2}$, where $d=2, 3, 4, 6$ [Finite Fields Appl. 35 (2015) 215--230]. In this paper we concentrate our efforts on the PPs of more general form \[ f(x)=(ax^{q} +bx +c)^r ϕ((ax^{q} +bx +c)^{(q^2 -1)/d}) +ux^{q} +vx~~\text{over $\mathbb{F}_{q^2}$}, \] where $a,b,c,u,v \in \mathbb{F}_{q^2}$, $r \in \mathbb{Z}^{+}$, $ϕ(x)\in \mathbb{F}_{q^2}[x]$ and $d$ is an arbitrary positive divisor of $q^2-1$. The key step is the construction of a commutative diagram with specific properties, which is the basis of the Akbary--Ghioca--Wang (AGW) criterion. By employing the AGW criterion two times, we reduce the problem of determining whether $f(x)$ permutes $\mathbb{F}_{q^2}$ to that of verifying whether two more polynomials permute two subsets of $\mathbb{F}_{q^2}$. As a consequence, we find a series of simple conditions for $f(x)$ to be a PP of $\mathbb{F}_{q^2}$. These results unify and generalize some known classes of PPs.

math.NT↗
arXiv2018

A Note on Two Constructions of Zero-Difference Balanced Functions

Notes on two constructions of zero-difference balanced (ZDB) functions are made in this letter. Then ZDB functions over $\mathbb{Z}_{e}\times \prod_{i=0}^{k}{\mathbb{F}_{q_i}}$ are obtained. And it shows that all the known ZDB functions using cyclotomic cosets over $\mathbb{Z}_{n}$ are special cases of a generic construction. Moreover, applications of these ZDB functions are presented.

cs.IT↗