Finite Field Of Order Research Articles

On the cardinality of matrices with prescribed rank and partial trace over a finite field

Let F be the finite field of order q and M(n,r,F) be the set of n×n matrices of rank r over the field F. For α∈F and A∈M(n,F), letZA,rα={X∈M(n,r,F)|Tr(AX)=α}. In this article, we solve the problem of determining the cardinality of ZA,rα. We also solve the generalization of the problem to rectangular matrices.

Metrical properties of beta-transformations on power series rings

Let A ∞ be an integer ring of a formal Laurent series field in one variable 1 t over a finite field of order q, with a maximal ideal M ∞ . Then, we introduce beta-transformations U β on A ∞ that are topologically isomorphic and measurably isomorphic to the well-known beta-transformation T β on M ∞ . We prove various metrical properties of U β such as (total) ergodicity and mixing of any order through explicit representations of both U β and its nth iterate U β n with respect to the shift operators. Furthermore, we examine dynamical connections between the measure-preserving property of 1-Lipschitz functions and the locally scaling property of ( q − k , q k ) -Lipschitz functions, in terms of the coefficients of van der Put and Carlitz–Wagner.

Unit Group of the Group Algebra $\mathbb{F}_qGL(2,7)$

Unit Group of the Group Algebra $\mathbb{F}_qGL(2,7)$

XOR count and block circulant MDS matrices over finite commutative rings

<p>Block circulant MDS matrices are used in the design of linear diffusion layers for lightweight cryptographic applications. Most of the work on construction of block circulant MDS matrices focused either on finite fields or $ GL(m, \mathbb{F}_2) $. The main objective of this paper is to extend the above study of block circulant MDS matrices to finite commutative rings. Additionally, we examine the behavior of the XOR count distribution under different reducible polynomials of equal degree over $ \mathbb{F}_2 $. We show that the determinant of a block circulant matrix over a ring can be expressed in a simple form. We construct $ 4 \times 4 $ and $ 8 \times 8 $ block circulant matrices over a ring. Furthermore, for non-negative integer $ l $, we identify the conditions under which a ring $ \mathfrak{R}_l = \frac{\mathbb{F}_2[x]}{\langle (f(x))^{2^l} \rangle} $, contains a finite field of order $ 2^m $, where $ f(x) $ is an irreducible polynomial of degree $ m $. To facilitate efficient implementation, we analyze XOR distributions within specific rings, such as $ R_1 = \frac{\mathbb{F}_2[x]}{\langle (1+x^2+x^6) \rangle} $ and $ R_2 = \frac{\mathbb{F}_2[x]}{\langle (1+x^4+x^6) \rangle} $. Our calculations reveal distinct XOR distributions when utilizing two reducible polynomials of equal degree, with XOR count distributions 776 and 764, respectively. However, when using irreducible polynomials of the same degree, the XOR count distributions remain the same. This difference is advantageous for applications in lightweight cryptography.</p>

Classification of some permutation quadrinomials from self reciprocal polynomials over [formula omitted

Let Fq be a finite field of order q. In this paper, for q even, we give a simple way to construct small degree polynomials over Fq with no roots in the set μq+1={a∈Fq2|aq+1=1} of the (q+1)-th roots of unity. Further, we use these small degree polynomials over Fq to construct and classify certain new families of permutation quadrinomials over Fq2.

Distribution of similar configurations in subsets of [formula omitted

Let Fq be a finite field of order q and E be a set in Fqd. The distance set of E is defined by Δ(E):={‖x−y‖:x,y∈E}, where ‖α‖=α12+…+αd2. Iosevich, Koh and Parshall (2018) proved that if d≥2 is even and |E|≥9qd/2, thenFq=Δ(E)Δ(E)={ab:a∈Δ(E),b∈Δ(E)∖{0}}. In other words, for each r∈Fq⁎ there exist (x,y)∈E2 and (x′,y′)∈E2 such that ‖x−y‖≠0 and ‖x′−y′‖=r‖x−y‖.Geometrically, this means that if the size of E is large, then for any given r∈Fq⁎ we can find a pair of edges in the complete graph K|E| with vertex set E such that one of them is dilated by r∈Fq⁎ with respect to the other. A natural question arises whether it is possible to generalize this result to arbitrary subgraphs of K|E| with vertex set E and this is the goal of this paper.In this paper, we solve this problem for k-paths (k≥2), simplexes and 4-cycles. We are using a mix of tools from different areas such as enumerative combinatorics, group actions and Turán type theorems.

Structural theorems on the distance sets over finite fields

Abstract Let F q \mathbb{F}_{q} be a finite field of order 𝑞. Iosevich and Rudnev [Erdős distance problem in vector spaces over finite fields, Trans. Amer. Math. Soc. 359 (2007), 12, 6127–6142] proved that, for any set A ⊂ F q d A\subset\mathbb{F}_{q}^{d} , if | A | ≫ q d + 1 2 \lvert A\rvert\gg q^{\frac{d+1}{2}} , then the distance set Δ ⁢ ( A ) \Delta(A) contains a positive proportion of all distances. Although this result is sharp in odd dimensions, it is conjectured that the right exponent should be d 2 \frac{d}{2} in even dimensions. During the last 15 years, only some improvements have been made in two dimensions, and the conjecture is still wide open in higher dimensions. To fill the gap, we need to understand more about the structures of the distance sets; the main purpose of this paper is to provide some structural theorems on the distribution of square and non-square distances.

On Galois groups of linearized polynomials related to the special linear group of prime degree

Let F be any field of prime characteristic p and let q be a power of p. We assume that F contains the finite field of order q. A q-polynomial L over F is an element of the polynomial ring F[x] with the property that all exponents are a power of q. We assume that the coefficient of the x term of L is nonzero. We investigate the Galois group G of L over F, under the assumption that L(x)/x is irreducible in F[x]. It is well known that if L has q-degree n, its roots are an n-dimensional vector space over the field of order q and G acts linearly on this space.Our main theorem is the following. We consider a monic q-polynomial L whose q-degree is a prime, r, say, with L(x)/x irreducible over F. Assuming that the coefficient of the x term of L is (−1)r, we show that the Galois group of L over F must be the special linear group SL(r,q) of degree r over the field of order q when q>2. We also prove a projective version for the projective special linear group PSL(r,q), and we present the analysis when q=2.

A New Construction of Strongly Regular Graphs with Parameters of the Complement Symplectic Graph

The symplectic graph $Sp(2d, q)$ is the collinearity graph of the symplectic space of dimension $2d$ over the finite field of order $q$. A $k$-regular graph on $v$ vertices is a divisible design graph with parameters $(v,k,\lambda_1,\lambda_2,m,n)$ if its vertex set can be partitioned into $m$ classes of size $n$, such that any two different vertices from the same class have $\lambda_1$ common neighbours, and any two vertices from different classes have $\lambda_2$ common neighbours whenever it is not complete or edgeless. In this paper we propose a new construction of strongly regular graphs with the parameters of the complement of the symplectic graph using divisible design graphs.

Configurations of rectangles in [formula omitted

Let Fq be a finite field of order q. In this paper, we study the distribution of rectangles in a given set in Fq2. More precisely, for any 0<δ≤1, we prove that there exists an integer q0=q0(δ) with the following property: if q≥q0 and A is a multiplicative subgroup of Fq⁎ with |A|≥q2/3, then any set S⊂Fq2 with |S|≥δq2 contains at least ≫|S|4|A|2q5 rectangles with side-lengths in A. We also consider the case of rectangles with one fixed side-length and the other in a multiplicative subgroup A.

Counting formulas for certain p-subgroups of GL_n(F_p)

Let p be a prime number and F_p a finite field of order p. Let GL_n(F_p) denote the general linear group and let U_n denote the unitriangular group of n x n upper triangular matrices with ones on the diagonal, over the finite field F_p. This is a finite group of order and a Sylow p-subgroup of GL_n(F_p}. In this work, we characterize some p-subgroups of GL_n(F_p) with respect to a given property.

Weight enumerators of all irreducible cyclic codes of length n

Let [Formula: see text], where [Formula: see text] are distinct odd primes, be an integer and [Formula: see text] be a finite field of order [Formula: see text] with [Formula: see text]. We determine the weight enumerators of all irreducible cyclic codes of length [Formula: see text] over [Formula: see text] when multiplicative order of [Formula: see text] modulo [Formula: see text] is [Formula: see text]; [Formula: see text] and [Formula: see text]; [Formula: see text], where [Formula: see text].

Large Orbits on Markoff-Type K3 Surfaces over Finite Fields

Abstract We study the surface $\mathcal {W}_k: x^2 + y^2 + z^2 + x^2 y^2 z^2 = k x y z$ in $(\mathbb {P}^1)^3$, a tri-involutive K3 (TIK3) surface. We explain a phenomenon noticed by Fuchs, Litman, Silverman, and Tran: over a finite field of order $\equiv 1$ mod $8$, the points of $\mathcal {W}_4$ do not form a single large orbit under the group $\Gamma $ generated by the three involutions fixing two variables and a few other obvious symmetries, but rather admit a partition into two $\Gamma $-invariant subsets of roughly equal size. The phenomenon is traced to an explicit double cover of the surface.

Jordan Types of Triangular Matrices over a Finite Field

Let \(\lambda \) be a partition of an integer n and \({\mathbb F}_q\) be a finite field of order q. Let \(P_\lambda (q)\) be the number of strictly upper triangular \(n\times n\) matrices of the Jordan type \(\lambda \). It is known that the polynomial \(P_\lambda \) has a tendency to be divisible by high powers of q and \(Q=q-1\), and we put \(P_\lambda (q)=q^{d(\lambda )}Q^{e(\lambda )}R_\lambda (q)\), where \(R_\lambda (0)\ne 0\) and \(R_\lambda (1)\ne 0\). In this article, we study the polynomials \(P_\lambda (q)\) and \(R_\lambda (q)\). Our main results: an explicit formula for \(d(\lambda )\) (an explicit formula for \(e(\lambda )\) is known, see Proposition 3.3 below), a recursive formula for \(R_\lambda (q)\) (a similar formula for \(P_\lambda (q)\) is known, see Proposition 3.1 below), the stabilization of \(R_\lambda \) with respect to extending \(\lambda \) by adding strings of 1’s, and an explicit formula for the limit series \(R_{\lambda 1^\infty }\). Our studies are motivated by projected applications to the orbit method in the representation theory of nilpotent algebraic groups over finite fields.

Unitary inner product graphs and their automorphisms

Let [Formula: see text] be a finite field of order [Formula: see text] and [Formula: see text] be an integer with [Formula: see text] or [Formula: see text], where [Formula: see text] is a power of a prime. We introduce the concept of the unitary inner product graph [Formula: see text] over [Formula: see text] and determine its automorphism group. We obtain two necessary and sufficient conditions for two vertices of [Formula: see text] and two edges of [Formula: see text], respectively, are in the same orbit under the action of the automorphism group of [Formula: see text]

Solids in the space of the Veronese surface in even characteristic

This paper forms part of a project aiming to determine the orbits of subspaces of PG(5,q) under the subgroup PGL(3,q) of the setwise stabiliser of the Veronese surface in PGL(6,q), where Fq is the finite field of order q. Specifically, we classify the orbits of solids of PG(5,q) in the case where q is even. We also determine two useful combinatorial invariants of each type of solid, namely their point-orbit and hyperplane-orbit distributions. Additionally, we calculate the stabiliser in PGL(3,q) of each (type of) solid S, and thereby determine the size of each orbit. The classification of solids is equivalent to the classification of pencils of conics in PG(2,q), q even, under coordinate transformations. The latter classification is known, but it seems that no proof is recorded in the literature, which generally points to a 1927 article of Campbell containing only an incomplete classification. Our approach is different and independent of Campbell's work, which we correct and complete.

More results on hulls of some primitive binary and ternary BCH codes

The (Euclidean) hull of a linear code is defined to be the intersection of the code and its Euclidean dual. It is clear that the hulls are self-orthogonal codes, which are an important type of linear codes due to their wide applications in communication and cryptography. Let C be an [n,k] cyclic code over Fq, where Fq is the finite field of order q. In this paper, we will employ the defining set of the code C to present a general characterization when its hull has dimension k−ℓ. Furthermore, we mainly focus on the primitive q-ary BCH codes C(q,n,δ,b) when b=0 and b=1 based on the general characterization. Especially for binary and ternary cases, we will present several sufficient and necessary conditions that the hulls of the codes C(q,n,δ,b) have dimensions k−2 and k−3 by giving lower and upper bounds on their designed distances, which extends the results of [17]. In addition, several classes of binary and ternary self-orthogonal codes are proposed via the hulls of BCH codes and their parameters are investigated in some special cases.

Classication of Units of Five Radical Zero Completely Primary Finite Rings with Variant Orders of Second Galois Ring Module Generators

Let \(R\) be a commutative completely primary finite ring with a unique maximal ideal \(Z(R)\) such that \((Z(R))^{5}=(0) ;(Z(R))^{4} \neq(0)\). Then \(R / Z(R) \cong G F\left(p^{r}\right)\) is a finite field of order \(p^{r}\). Let \(R_{0}=G R\left(p^{k r}, p^{k}\right)\) be a Galois ring of order \(p^{k r}\) and of characteristic \(p^{k}\) for some prime number \(p\) and positive integers \(k, r\) so that \(R=R_{0} \oplus U \bigoplus V \bigoplus W \bigoplus Y\), where \(U, V, W\) and \(Y\) are \(R_{0} / p R_{0}\) - spaces considered as \(R_{0}\) modules generated by \(e, f, g\) and \(h\) elements respectively. Then \(R\) is of characteristic \(p^{k}\) where \(1 \leq k \leq 5\). In this paper, we investigate and determine the structures of the unit groups of some classes of commutative completely primary finite ring \(R\) with \(p u_{i}=p^{\xi} v_{j}=p w_{k}=p y_{l}=0\), where \(\xi=2,3 ; 1 \leq i \leq e, 1 \leq j \leq f, 1 \leq k \leq g\), and \(1 \leq l \leq h\).

Explicit factorization of and a-constacyclic codes over a finite field

Let be a finite field of order q, t be a prime and be positive integers. In this article, we find all irreducible divisors of over where and such that are distinct odd primes and c is a positive integer with and Moreover, we construct an a-constacyclic code by using these irreducible divisors.

Rédei permutations with the same cycle structure

Let Fq be the finite field of order q, and P1(Fq)=Fq∪{∞}. Write (x+y)m as N(x,y)+D(x,y)y. For m∈N and a∈Fq, the Rédei function Rm,a:P1(Fq)→P1(Fq) is defined by N(x,a)/D(x,a) if D(x,a)≠0 and x≠∞, and ∞, otherwise. In this paper we give a complete characterization of all pairs (m,n)∈N2 such that the Rédei permutations Rm,a and Rn,b have the same cycle structure when a and b have the same quadratic character and q is odd. We explore some relationships between such pairs (m,n), and provide explicit families of Rédei permutations with the same cycle structure. When a Rédei permutation has a unique cycle structure that is not shared by any other Rédei permutation, we call it isolated. We show that the only isolated Rédei permutations are the isolated Rédei involutions. Moreover, all our results can be transferred to bijections of the form mx and xm on certain domains.