On Existence of Primitive Normal Elements of Cubic Form over Finite Fields

For a prime [Formula: see text]and a positive integer[Formula: see text], let [Formula: see text] and [Formula: see text] be the extension field of [Formula: see text]. We derive a sufficient condition for the existence of a primitive element [Formula: see text] in[Formula: see text] such that [Formula: see text] is also a primitive element of [Formula: see text], a sufficient condition for the existence of a primitive normal element [Formula: see text] in [Formula: see text] over [Formula: see text] such that [Formula: see text] is a primitive element of [Formula: see text], and a sufficient condition for the existence of a primitive normal element [Formula: see text] in [Formula: see text] over [Formula: see text] such that [Formula: see text] is also a primitive normal element of [Formula: see text] over [Formula: see text].

Minimal degrees of algebraic numbers with respect to primitive elements

Given a number field [Formula: see text], we define the degree of an algebraic number [Formula: see text] with respect to a choice of a primitive element of [Formula: see text]. We propose the question of computing the minimal degrees of algebraic numbers in [Formula: see text], and examine these values in degree 4 Galois extensions over [Formula: see text] and triquadratic number fields. We show that computing minimal degrees of non-rational elements in triquadratic number fields is closely related to solving classical Diophantine problems such as congruent number problem as well as understanding various arithmetic properties of elliptic curves.

A Modeling Method of Cylindrical Turning Processing Behavior

This paper put forward the theory of processing behavior through research on conceptual system of advanced manufacturing technologies and production modes such as cloud manufacturing and systematic classification of processing technology, and gave the concept of processing behavior primitive element. The processing behavior was classified and the relationship between different levels of processing behavior was clarified. To realize the standardization of NC processing technology for parts of automatic turning programming. Embed turning behavior into the design elements, realize the combination of design and manufacturing, shorten the time and space of coordination and product cycle, reduce product design and manufacturing costs, and achieve green, low consumption, high quality, high efficiency and effective manufacturing of turning parts.

Repeated-root constacyclic codes of length 6lmpn

<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{F}_{q} $\end{document}</tex-math></inline-formula> be a finite field with character <inline-formula><tex-math id="M2">\begin{document}$ p $\end{document}</tex-math></inline-formula>. In this paper, the multiplicative group <inline-formula><tex-math id="M3">\begin{document}$ \mathbb{F}_{q}^{*} = \mathbb{F}_{q}\setminus\{0\} $\end{document}</tex-math></inline-formula> is decomposed into a mutually disjoint union of <inline-formula><tex-math id="M4">\begin{document}$ \gcd(6l^mp^n,q-1) $\end{document}</tex-math></inline-formula> cosets over subgroup <inline-formula><tex-math id="M5">\begin{document}$ &lt;\xi^{6l^mp^n}&gt; $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M6">\begin{document}$ \xi $\end{document}</tex-math></inline-formula> is a primitive element of <inline-formula><tex-math id="M7">\begin{document}$ \mathbb{F}_{q} $\end{document}</tex-math></inline-formula>. Based on the decomposition, the structure of constacyclic codes of length <inline-formula><tex-math id="M8">\begin{document}$ 6l^mp^n $\end{document}</tex-math></inline-formula> over finite field <inline-formula><tex-math id="M9">\begin{document}$ \mathbb{F}_{q} $\end{document}</tex-math></inline-formula> and their duals is established in terms of their generator polynomials, where <inline-formula><tex-math id="M10">\begin{document}$ p\neq{3} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M11">\begin{document}$ l\neq{3} $\end{document}</tex-math></inline-formula> are distinct odd primes, <inline-formula><tex-math id="M12">\begin{document}$ m $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M13">\begin{document}$ n $\end{document}</tex-math></inline-formula> are positive integers. In addition, we determine the characterization and enumeration of all linear complementary dual(LCD) negacyclic codes and self-dual constacyclic codes of length <inline-formula><tex-math id="M14">\begin{document}$ 6l^mp^n $\end{document}</tex-math></inline-formula> over <inline-formula><tex-math id="M15">\begin{document}$ \mathbb{F}_{q} $\end{document}</tex-math></inline-formula>.</p>

The existence of primitive normal elements of quadratic forms over finite fields

For [Formula: see text] ([Formula: see text]), denote by [Formula: see text] the finite field of order [Formula: see text] and for a positive integer [Formula: see text], let [Formula: see text] be its extension field of degree [Formula: see text]. We establish a sufficient condition for existence of a primitive normal element [Formula: see text] such that [Formula: see text] is a primitive element, where [Formula: see text], with [Formula: see text] satisfying [Formula: see text] in [Formula: see text].

Separating singular moduli and the primitive element problem

We prove that ${|x-y|\ge 800X^{-4}}$, where $x$ and $y$ are distinct singular moduli of discriminants not exceeding $X$. We apply this result to the ‘primitive element problem’ for two singular moduli. In a previous article, Faye and Riffaut show that the number field ${{\mathbb{Q}}}(x,y)$, generated by two distinct singular moduli $x$ and $y$, is generated by ${x-y}$ and, with some exceptions, by ${x+y}$ as well. In this article we fix a rational number ${\alpha \ne 0,\pm 1}$ and show that the field ${{\mathbb{Q}}}(x,y)$ is generated by ${x+\alpha y}$, with a few exceptions occurring when $x$ and $y$ generate the same quadratic field over ${{\mathbb{Q}}}$. Together with the above-mentioned result of Faye and Riffaut, this generalizes a theorem due to Allombert et al. (2015) about solutions of linear equations in singular moduli.

Primitive element pairs with a prescribed trace in the quartic extension of a finite field

In this paper, we give a largely self-contained proof that the quartic extension [Formula: see text] of the finite field [Formula: see text] contains a primitive element [Formula: see text] such that the element [Formula: see text] is also a primitive element of [Formula: see text] and [Formula: see text] for any prescribed [Formula: see text]. The corresponding result has already been established for finite field extensions of degrees exceeding 4 in [Primitive element pairs with one prescribed trace over a finite field, Finite Fields Appl. 54 (2018) 1–14.].

New Constructions of Balanced Boolean Functions with Maximum Algebraic Immunity, High Nonlinearity and Optimal Algebraic Degree

This paper consists of proposal of two new constructions of balanced Boolean function achieving a new lower bound of nonlinearity along with high algebraic degree and optimal or highest algebraic immunity. This construction has been made by using representation of Boolean function with primitive elements. Galois Field, used in this representation has been constructed by using powers of primitive element such that greatest common divisor of power and is 1. The constructed balanced variable Boolean functions achieve higher nonlinearity, algebraic degree of , and algebraic immunity of for odd , for even . The nonlinearity of Boolean function obtained in the proposed constructions is better as compared to existing Boolean functions available in the literature without adversely affecting other properties such as balancedness, algebraic degree and algebraic immunity.