Finite Field Fq Research Articles

Smooth symmetric systems over a finite field and applications

We study the set of common Fq–rational solutions of “smooth” systems of multivariate symmetric polynomials with coefficients in a finite field Fq. We show that, under certain conditions, the set of common solutions of such polynomial systems over the algebraic closure of Fq has a “good” geometric behavior. This allows us to obtain precise estimates on the corresponding number of common Fq–rational solutions. In the case of hypersurfaces we are able to strengthen the results. We illustrate the interest of these estimates through their application to certain classical combinatorial problems over finite fields.

New upper bounds on the number of non-zero weights of constacyclic codes

For any simple-root constacyclic code C over a finite field Fq, as far as we know, the group G generated by the multiplier, the constacyclic shift and the scalar multiplications is the largest subgroup of the automorphism group Aut(C) of C. In this paper, by calculating the number of G-orbits of C﹨{0}, we give an explicit upper bound on the number of non-zero weights of C and present a necessary and sufficient condition for C to meet the upper bound. Some examples in this paper show that our upper bound is tight and better than the upper bounds in Zhang and Cao (2024) [26]. In particular, our main results provide a new method to construct few-weight constacyclic codes. Furthermore, for the constacyclic code C belonging to two special types, we obtain a smaller upper bound on the number of non-zero weights of C by substituting G with a larger subgroup of Aut(C). The results derived in this paper generalize the main results in Chen et al. (2024) [9].

On parallelisms of PG(3,4) with automorphisms of order 2

Let PG(n,q) be the n-dimensional projective space over the finite field Fq. A spread in PG(n,q) is a set of mutually skew lines which partition the point set. A parallelism is a partition of the set of lines by spreads. The classification of parallelisms in small finite projective spaces is of interest for problems from projective geometry, design theory, network coding, error-correcting codes, and cryptography. All parallelisms of PG(3,2) and PG(3,3) are known. Parallelisms of PG(3,4) which are invariant under automorphisms of odd prime orders have also been classified. The present paper contributes to the classification of parallelisms in PG(3,4) with automorphisms of even order. We focus on cyclic groups of order four and the group of order two generated by a Baer involution. We examine invariants of the parallelisms such as the full automorphism group, the type of spreads and questions of duality. The results given in this paper show that the number of parallelisms of PG(3,4) is at least 8675365. Some future directions of research are outlined.

Two classes of LCD BCH codes over finite fields

BCH codes form a special subclass of cyclic codes and have been extensively studied in the past decades. Determining the parameters of BCH codes, however, has been an important but difficult problem. Recently, in order to further investigate the dual codes of BCH codes, the concept of dually-BCH codes was proposed. In this paper, we study BCH codes of lengths qm+1q+1 and qm+1 over the finite field Fq, both of which are LCD codes. The dimensions of narrow-sense BCH codes of length qm+1q+1 with designed distance δ=ℓqm−12+1 are determined, where q>2 and 2≤ℓ≤q−1. Lower bounds on the minimum distances of the dual codes of narrow-sense BCH codes of length qm+1 are developed for odd q, which are good in some cases. Moreover, sufficient and necessary conditions for the even-like subcodes of narrow-sense BCH codes of length qm+1 being dually-BCH codes are presented, where q is odd and m≢0(mod4).

A Selberg Trace Formula for GL3(Fp)∖GL3(Fq)/K

In this paper, we prove a discrete analog of the Selberg Trace Formula for the group GL3(Fq). By considering a cubic extension of the finite field Fq, we define an analog of the upper half-space and an action of GL3(Fq) on it. To compute the orbital sums, we explicitly identify the double coset spaces and fundamental domains in our upper half space. To understand the spectral side of the trace formula, we decompose the induced representation ρ=IndΓG1 for G=GL3(Fq) and Γ=GL3(Fp).

Separating invariants for two-dimensional orthogonal groups over finite fields

We described a minimal separating set for the algebra of O2+(Fq)-invariant polynomial functions of m-tuples of two-dimensional vectors over a finite field Fq.

Irreducibility properties of Carlitz' binomial coefficients for algebraic function fields

We study the class of univariate polynomials βk(X), introduced by Carlitz, with coefficients in the algebraic function field Fq(t) over the finite field Fq with q elements. It is implicit in the work of Carlitz that these polynomials form an Fq[t]-module basis of the ring Int(Fq[t])={f∈Fq(t)[X]|f(Fq[t])⊆Fq[t]} of integer-valued polynomials on the polynomial ring Fq[t]. This stands in close analogy to the famous fact that a Z-module basis of the ring Int(Z) is given by the binomial polynomials (Xk).We prove, for k=qs, where s is a non-negative integer, that βk is irreducible in Int(Fq[t]) and that it is even absolutely irreducible, that is, all of its powers βkm with m>0 factor uniquely as products of irreducible elements of this ring. As we show, this result is optimal in the sense that βk is not even irreducible if k is not a power of q.

Arithmetic progression in a finite field with prescribed norms

Abstract Given a prime power q and a positive integer n, let 𝔽 q n {\mathbb{F}_{q^{n}}} represent a finite extension of degree n of the finite field 𝔽 q {{\mathbb{F}_{q}}} . In this article, we investigate the existence of m elements in arithmetic progression, where every element is primitive and at least one is normal with prescribed norms. Moreover, for n ≥ 6 {n\geq 6} , q = 3 k {q=3^{k}} , m = 2 {m=2} we establish that there are only 10 possible exceptions.

Higher reciprocity law and an analogue of the Grunwald–Wang theorem for the ring of polynomials over an ultra-finite field

In this paper, we establish an explicit higher reciprocity law for the polynomial ring over a nonprincipal ultraproduct of finite fields. Such an ultraproduct can be taken over the same finite field, which allows to recover the classical higher reciprocity law for the polynomial ring Fq[t] over a finite field Fq that is due to Dedekind, Kühne, Artin, and Schmidt. On the other hand, when the ultraproduct is taken over finite fields of unbounded cardinalities, we obtain an explicit higher reciprocity law for the polynomial ring over an infinite field in both characteristics 0 and p>0 for some prime p. We then use the higher reciprocity law to prove an analogue of the Grunwald–Wang theorem for such a polynomial ring in both characteristics 0 and p>0 for some prime p.

Equivalence for flag codes

Given a finite field Fq and a positive integer n, a flag is a sequence of nested Fq-subspaces of a vector space Fqn and a flag code is a nonempty collection of flags. The projected codes of a flag code are the constant dimension codes containing all the subspaces of prescribed dimensions that form the flags in the flag code.In this paper we address the notion of equivalence for flag codes and explore in which situations such an equivalence can be reduced to the equivalence of the corresponding projected codes. In addition, this study leads to new results concerning the automorphism group of certain families of flag codes, some of them also introduced in this paper.

Determinants of tridiagonal matrices over some commutative finite chain rings

Abstract Diagonal matrices and their generalization in terms of tridiagonal matrices have been of interest due to their nice algebraic properties and wide applications. In this article, the determinants of tridiagonal matrices over a finite field F q {{\mathbb{F}}}_{q} and a commutative finite chain ring R R are studied. The main focus is the enumeration of tridiagonal matrices with prescribed determinant. The number of tridiagonal matrices with prescribed determinant over F q {{\mathbb{F}}}_{q} and the number of non-singular tridiagonal matrices with prescribed determinant over R R are completely determined. For singular tridiagonal matrices with prescribed determinant over R R , bounds on the number of such matrices with prescribed determinant are given. Subsequently, the number of some special tridiagonal matrices with prescribed determinant over F q {{\mathbb{F}}}_{q} and R R is presented.

Determinants of Arrowhead Matrices over Finite Commutative Chain Rings

Arrowhead matrices have attracted attention due to their rich algebraic structures and numerous applications. In this paper, we focus on the enumeration of n × n arrowhead matrices with prescribed determinant over a finite field Fq and over a finite commutative chain ring R. The number of n × n arrowhead matrices over Fq of a fixed determinant a is determined for all positiveintegers n and for all elements a ∈ Fq. As applications, this result is used in the enumeration of n × n non-singular arrowhead matrices with prescribed determinant over R. Subsequently, some bounds on the number of n × n singular arrowhead matrices over R of a fixed determinant are given. Finally, some open problems are presented.

The quotient set of the quadratic distance set over finite fields

Abstract Let 𝔽 q d {\mathbb{F}_{q}^{d}} be the d-dimensional vector space over the finite field 𝔽 q {\mathbb{F}_{q}} with q elements. For each non-zero r in 𝔽 q {\mathbb{F}_{q}} and E ⊂ 𝔽 q d {E\subset\mathbb{F}_{q}^{d}} , we define W ⁢ ( r ) {W(r)} as the number of quadruples ( x , y , z , w ) ∈ E 4 {(x,y,z,w)\in E^{4}} such that Q ⁢ ( x - y ) Q ⁢ ( z - w ) = r {\frac{Q(x-y)}{Q(z-w)}=r} , where Q is a non-degenerate quadratic form in d variables over 𝔽 q {\mathbb{F}_{q}} . When Q ⁢ ( α ) = ∑ i = 1 d α i 2 {Q(\alpha)=\sum_{i=1}^{d}\alpha_{i}^{2}} with α = ( α 1 , … , α d ) ∈ 𝔽 q d {\alpha=(\alpha_{1},\ldots,\alpha_{d})\in\mathbb{F}_{q}^{d}} , Pham (2022) recently used the machinery of group actions and proved that if E ⊂ 𝔽 q 2 {E\subset\mathbb{F}_{q}^{2}} with q ≡ 3 ⁢ ( mod ⁡ 4 ) {q\equiv 3~{}(\operatorname{mod}\,4)} and | E | ≥ C ⁢ q {|E|\geq Cq} , then we have W ⁢ ( r ) ≥ c ⁢ | E | 4 q {W(r)\geq\frac{c|E|^{4}}{q}} for any non-zero square number r ∈ 𝔽 q {r\in\mathbb{F}_{q}} , where C is a sufficiently large constant, c is some number between 0 and 1, and | E | {|E|} denotes the cardinality of the set E. In this article, we improve and extend Pham’s result in two dimensions to arbitrary dimensions with general non-degenerate quadratic distances. As a corollary of our results, we also generalize the sharp results on the Falconer-type problem for the quotient set of distance set due to the first two authors and Parshall (2019). Furthermore, we provide improved constants for the size conditions of the underlying sets. The key new ingredient is to relate the estimate of the W ⁢ ( r ) {W(r)} to a quadratic homogeneous variety in 2 ⁢ d {2d} -dimensional vector space. This approach is fruitful because it allows us to take advantage of Gauss sums which are more handleable than the Kloosterman sums appearing in the standard distance-type problems.

A new approach to solving isomorphism problems of classical dynamical systems using algebraic structures

Abstract There exists a fixed rule in classical dynamical systems that describes a point in geometric space over time. In this paper, based on the algebraic structure perspective, the dynamical system is defined as a category characterized by ordered state projections, and the dynamical system is inscribed using the algebraic structure, covering the phase space, continuous self-maps containing a single parametric variable, and the dynamical system itself. Meanwhile, two types of self-isomorphisms of algebraic maps are explored. One is the self-isomorphism of ideal inclusion maps on an algebra Mn(Fq) consisting of full matrices of order n over a finite field Fq . The second is the self-isomorphism of ideal relational graphs on a finite field Fq . It is proved that any self-isomorphism problem of graph Mn (Fq ) when n >3 can be used with both criteria on it. Finally, a classical model of a dynamical system obtained from f(x) = cos x iterations is studied and its global convergence is discussed. Numerical experimental results show that the discrete dynamical system generated by function f(x) = cos x iteration has a unique ω limit point of 0.735, indicating that the stability and predictability of classical dynamical systems can be achieved using algebraic structures, as well as revealing the complexity, instability, and chaos of the system.

Minimal and maximal cyclic codes of length 2p

In this paper, we compute the maximal and minimal codes of length 2p over finite fields Fq with p and q are distinct odd primes and f(p) = p – 1 is the multiplicative order of q modulo 2p. We show that, every cyclic code is a direct sum of minimal cyclic codes.

Some relations between the irreducible polynomials over a finite field and its quadratic extension

In this paper, we establish some relations between irreducible polynomials over a finite field Fq and its quadratic extension Fq2. First we consider a relation between the numbers of irreducible polynomials of a fixed degree over Fq and Fq2, and some relations between self-reciprocal irreducible polynomials over Fq and self-conjugate-reciprocal irreducible polynomials over Fq2. We also obtain formulas for the number and the product of all self-conjugate-reciprocal irreducible monic (SCRIM) polynomials over Fq2.

On Ext1 for Drinfeld modules

Let A=Fq[t] be the polynomial ring over a finite field Fq and let ϕ and ψ be A-Drinfeld modules. In this paper we consider the group Ext1(ϕ,ψ) with the Baer addition. We show that if rankϕ>rankψ then Ext1(ϕ,ψ) has the structure of a t-module. We give complete algorithm describing this structure. We generalize this to the cases: Ext1(Φ,ψ) where Φ is a t-module and ψ is a Drinfeld module and Ext1(Φ,C⊗e) where Φ is a t-module and C⊗e is the e-th tensor product of Carlitz module. We also establish duality between Ext groups for t-modules and the corresponding adjoint tσ-modules. Finally, we prove the existence of “Hom−Ext” six-term exact sequences for t-modules and dual t-motives. As the category of t-modules is only additive (not abelian) this result is nontrivial.

On complete m-arcs

Let m be a positive integer and q be a prime power. For large finite base fields Fq, we show that any plane curve can be used to produce a complete m-arc as long as some generic explicit geometric conditions on the curve are verified. To show the effectiveness of our theory, we derive complete m-arcs from hyperelliptic curves and from Artin-Schreier curves.

Rota’s basis conjecture holds for random bases of vector spaces

In 1989, Rota conjectured that, given n bases B1,…,Bn of the vector space Fn over some field F, one can always decompose the multi-set B1∪⋯∪Bn into transversal bases. This conjecture remains wide open despite of a lot of attention. In this paper, we consider the setting of random bases B1,…,Bn. More specifically, our first result shows that Rota’s basis conjecture holds with probability 1−o(1) as n→∞ if the bases B1,…,Bn are chosen independently uniformly at random among all bases of Fqn for some finite field Fq (the analogous result is trivially true for an infinite field F). In other words, the conjecture is true for almost all choices of bases B1,…,Bn⊆Fqn. Our second, more general, result concerns random bases B1,…,Bn⊆Sn for some given finite subset S⊆F (in other words, bases B1,…,Bn where all vectors have entries in S). We show that when choosing bases B1,…,Bn⊆Sn independently uniformly at random among all bases that are subsets of Sn, then again Rota’s basis conjecture holds with probability 1−o(1) as n→∞.

On security of GIS systems with N-tier architecture and family of graph based ciphers

Discovery of q-regular tree description in terms of an infinite system of quadratic equations over finite field Fq had an impact on Computer Science, theory of graph based cryptographic algorithms in particular. It stimulates the development of new graph based stream ciphers. It turns out that such encryption instruments can be efficiently used in GIS protection systems which use N-Tier Architecture. We observe known encryption algorithms based on the approximations of regular tree, their modifications defined over arithmetical rings and implementations of these ciphers. Additionally new more secure graph based ciphers suitable for GIS protection will be presented.Algorithms are constructed using vertices of bipartite regular graphs D(n,K) defined by a finite commutative ring K with a unit and a non-trivial multiplicative group K*. The partition of such graphs are n-dimensional affine spaces over the ring K. A walk of even length determines the transformation of the transition from the initial to the last vertex from one of the partitions of the graph. Therefore, the affine space Kn can be considered as a space of plaintexts, and walking on the graph is a password that defines the encryption transformation.With certain restrictions on the password the effect when different passwords with K*)2s, s <[(n+5)/2]/2 correspond to different ciphertexts of the selected plaintext with Kn can be achieved.In 2005, such an algorithm in the case of the finite field F127 was implemented for the GIS protection. Since then, the properties of encryption algorithms using D(n, K) graphs (execution speed, mixing properties, degree and density of the polynomial encryption transform) have been thoroughly investigated.The complexity of linearization attacks was evaluated and modifications of these algorithms with the resistance to linearization attacks were found. It turned out that together with D(n, K) graphs, other algebraic graphs with similar properties, such as A(n, K) graphs, can be effectively used.The article considers several solutions to the problem of protecting the geological information system from possible cyberattacks using stream ciphers based on graphs. They have significant advantages compared to the implemented earlier systems.