Related papers: A new trace bilinear form on cyclic $\mathbb{F}_q$…

For a prime power \( q \) and a positive integer \( n \), a subspace \( U \subseteq \mathbb{F}_q^n \) is called cyclically covering if the union of all its cyclic shifts covers the whole space \( \mathbb{F}_q^n \). Let \( h_q(n) \) denote…

Let $n\ge 2$ be an integer and let $\mathbb F_q$ be the finite field with $q$ elements, where $q$ is a prime power. Given $\mathbb F_q$-affine hyperplanes $\mathcal A_1, \ldots, \mathcal A_n$ of $\mathbb F_{q^n}$ in general position, we…

Cyclic and self-dual codes are important classes of codes in coding theory. Jia, Ling and Xing \cite{Jia} as well as Kai and Zhu \cite{Kai} proved that Euclidean self-dual cyclic codes of length $n$ over $\mathbb{F}_q$ exist if and only if…

We propose a construction of lattices from (skew-) polynomial codes, by endowing quotients of some ideals in both number fields and cyclic algebras with a suitable trace form. We give criteria for unimodularity. This yields integral and…

Though it is well known that the roots of any affine polynomial over a finite field can be computed by a system of linear equations by using a normal base of the field, such solving approach appears to be difficult to apply when the field…

We present several existence and nonexistence results for permutation binomials of the form $x^r(x^{q-1}+a)$, where $e\geq 2$ and $a\in \mathbb{F}_{q^e}^*$. As a consequence, we obtain a complete characterization of such permutation…

Long quasi-cyclic codes of any fixed index $>1$ have been shown to be asymptotically good, depending on Artin primitive root conjecture in (A. Alahmadi, C. G\"uneri, H. Shoaib, P. Sol\'e, 2017). We use this recent result to construct good…

The tensor product of a positive and a negative discrete series representation of the quantum algebra U_q(su(1,1)) decomposes as a direct integral over the principal unitary series representations. Discrete terms can appear, and these terms…

In the mid 80's Conner and Perlis showed that for cyclic number fields of prime degree $p$ the isometry class of integral trace is completely determined by the discriminant. Here we generalize their result to tame cyclic number fields of…

In this paper, we connect two types of representations of a permutation $\sigma$ of the finite field $\F_q$. One type is algebraic, in which the permutation is represented as the composition of degree-one polynomials and $k$ copies of…

We define a family of quaternary sequences over the residue class ring modulo $4$ of length $pq$, a product of two distinct odd primes, using the generalized cyclotomic classes modulo $pq$ and calculate the discrete Fourier transform (DFT)…

We prove that for any prime power $q\notin\{3,4,5\}$, the cubic extension $\mathbb{F}_{q^3}$ of the finite field $\mathbb{F}_q$ contains a primitive element $\xi$ such that $\xi+\xi^{-1}$ is also primitive, and…

In a recent paper, Tang and Ding introduced a class of binary cyclic codes of rate close to one half with a designed lower bound on their minimum distance. The definition involves the base $2$ expansion of the integers in their defining…

The Factorial Basis method, initially designed for quasi-triangular, shift-compatible factorial bases, provides solutions to linear recurrence equations in the form of definite-sums. This paper extends the Factorial Basis method to its…

Let $R=\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}+\cdots+u^{k}\mathbb{F}_{2^{m}}$ , where $\mathbb{F}_{2^{m}}$ is a finite field with $2^{m}$ elements, $m$ is a positive integer, $u$ is an indeterminate with $u^{k+1}=0.$ In this paper, we…

Let $\mathbb{F}_q$ denote the finite fields with $q$ elements. The permutation behavior of several classes of infinite families of permutation polynomials over finite fields have been studied in recent years. In this paper, we continue with…

The Fibonomial coefficients are well-known analogues of the classical binomial coefficients. In 2009, Sagan and Savage introduced a combinatorial interpretation for these coefficients, based on tiling a rectangular grid. More recently,…

Quasi-twisted (QT) codes are a generalization of quasi-cyclic (QC) codes. Based on consta-cyclic simplex codes, a new explicit construction of a family of 2-generator quasi-twisted (QT) two-weight codes is presented. It is also shown that…

Given a prime power $q$ and a positive integer $n$, let $\mathbb{F}_{q^{n}}$ denote the finite field with $q^n$ elements. Also let $a,b$ be arbitrary members of the ground field $\mathbb{F}_{q}$. We investigate the existence of a non-zero…

Let $K=\mathbb{Q}(\sqrt[n]{a})$ be an extension of degree $n$ of the field $\Q$ of rational numbers, where the integer $a$ is such that for each prime $p$ dividing $n$ either $p\nmid a$ or the highest power of $p$ dividing $a$ is coprime to…