In the study of (partial) difference sets and their generalizations in groups $G$, the most widely used method is to translate their definition into an equation over group ring $\mathbb{Z}[G]$ and to investigate this equation by applying complex representations of $G.$ In this paper, we investigate the existence of (partial) difference sets in a different way. We project the group ring equations in $\mathbb{Z}[G]$ to $\mathbb{Z}[N]$ where $N$ is a quotient group of $G$ isomorphic to the additive group of a finite field, and then use polynomials over this finite field to derive some existence conditions.

Let $G$ be a finite abelian group, $M$ a set of integers and $S$ a subset of $G$. We say that $M$ and $S$ form a splitting of $G$ if every nonzero element $g$ of $G$ has a unique representation of the form $g=ms$ with $m\in M$ and $s\in S$, while 0 has no such representation. The splitting is called purely singular if for each prime divisor $p$ of $|G|$, there is at least one element of $M$ is divisible by $p$. In this paper, we continue the study of purely singular splittings of cyclic groups. We prove that if $k\geq 2$ is a positive integer such that $[−2k+1, 2k+2]^∗$ splits a cyclic group $\mathbb{Z}_m$, then $m=4k+2$. We prove also that if $M=[−k_1, k_2]^∗$ splits $\mathbb{Z}_m$ purely singularly, and $15 \leq k_1+k_2 \leq 30$, then $m = 1$, or $m = k_1+k_2+1$, or $k_1 = 0$ and $m=2k_2+1$.

Let $\mathbb{F}_{p^m}$ be a finite field with $p^m$ elements, where $p$ is an odd prime and $m$ is a positive integer. Recently, [17] and [35] determined the weight distributions of subfield codes with the form $$\mathcal{C}_f=\{(({\rm Tr}(af(x)+bx)+c)_{x\in \mathbb{F}_{p^m}},{\rm Tr}(a)):a,b\in \mathbb{F}_{p^m},c\in \mathbb{F}_p\}$$ for $f(x) = x^2$ and $f(x) = x^{p^k+1}$ , respectively, where $Tr(·)$ is the trace function from $\mathbb{F}_{p^m}$ to $\mathbb{F}_p$, and $k$ is a nonnegative integer. In this paper, we further investigate the subfield code $\mathcal{C}_f$ for $f(x)$ being a known perfect nonlinear function over $\mathbb{F}_{p^m}$ and generalize some results in [17, 35]. The weight distributions of the constructed codes are determined by applying the theory of quadratic forms and the properties of perfect nonlinear functions over finite fields. In addition, the parameters of the duals of these codes are also determined. Several examples show that some of our codes and their duals have the best known parameters according to the code tables in [16]. The duals of some proposed codes are optimal according to the Sphere Packing bound if $p\geq 5$.

Let $\mathbb{F}_q$ be a finite field and $\mathbb{F}_{q^s}$ be an extension of $\mathbb{F}_q$. Let $f(x)\in \mathbb{F}_q[x]$ be a polynomial of degree $n$ with ${\rm gcd}(n,q) = 1$. We present a recursive formula for evaluating the exponential sum $\sum_{c\in \mathbb{F}_{q^s}} \chi^{(s)}(f(x))$. Let $a$ and $b$ be two elements in $\mathbb{F}_q$ with a $a≠0$, $u$ be a positive integer. We obtain an estimate for the exponential sum $\sum_{c\in \mathbb{F}^∗_{q^s}}\chi^{(s)} (ac^u+bc^{−1})$, where $\chi^{(s)}$ is the lifting of an additive character $\chi$ of $\mathbb{F}_q$. Some properties of the sequences constructed from these exponential sums are provided too.

In this paper, we first propose the maximum arc problem, normal rational curve conjecture, and extensions of normal rational curves over finite local rings, analogously to the finite geometry over finite fields. We then study the deep hole problem of generalized Reed-Solomon (RS) codes over finite local rings. Several different classes of deep holes are constructed. The relationship between finite geometry and deep holes of RS codes over finite local rings are also studied.

Permutation polynomials with low differential uniformity and high nonlinearity are preferred in cryptographic systems. In 2018, Tu, Zeng and Helleseth constructed a new class of permutation quadrinomials over the finite field $\mathbb{F}_{2^{2m}}$ for an odd integer $m$. In this paper, we aim to investigate the differential uniformity and nonlinearity of this class of permutation polynomials so as to find 4-uniform permutation polynomials with high nonlinearity.

This paper is concerned with so-called index $d$ generalized cyclotomic mappings of a finite field $\mathbb{F}_q$, which are functions $\mathbb{F}_q \rightarrow \mathbb{F}_q$ that agree with a suitable monomial function $x\mapsto ax^r$ on each coset of the index $d$ subgroup of $\mathbb{F}^∗_q$. We discuss two important rewriting procedures in the context of generalized cyclotomic mappings and present applications thereof that concern index $d$ generalized cyclotomic permutations of $\mathbb{F}_q$ and pertain to cycle structures, the classification of $(q−1)$-cycles and involutions, as well as inversion.

We introduce a new infinite class of bipartite graphs, called jumped Wenger graphs, which has a similar structure with Wenger graphs. We give a tight upper bound of the diameter for these graphs and the exact diameter for some special graphs. We also determine the girth of the jumped Wenger graphs.