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

We explicitly find regulators of an infinite family {Lm} of the simplest quartic function fields with a parameter m in a polynomial ring [t], where is the finite field of order q with odd characteristic. In fact, this infinite family of the simplest quartic function fields are subfields of maximal real subfields of cyclotomic function fields having the same conductors. We obtain a lower bound on the class numbers of the family {Lm} and some result on the divisibility of the divisor class numbers of cyclotomic function fields that contain {Lm} as their subfields. Furthermore, we find an explicit criterion for the characterization of splitting types of all the primes of the rational function field (t) in {Lm}.

Schoof’s algorithm computes the number m m of points on an elliptic curve E E defined over a finite field F q {\Bbb F}_q . Schoof determines m m modulo small primes ℓ \ell using the characteristic equation of the Frobenius of E E and polynomials of degree O ( ℓ 2 ) O(\ell ^2) . With the works of Elkies and Atkin, we have just to compute, when ℓ \ell is a “good" prime, an eigenvalue of the Frobenius using polynomials of degree O ( ℓ ) O(\ell ) . In this article, we compute the complexity of Müller’s algorithm, which is the best known method for determining one eigenvalue and we improve the final step in some cases. Finally, when ℓ \ell is “bad", we describe how to have polynomials of small degree and how to perform computations, in Schoof’s algorithm, on x x -values only.

The computation of the order of Frobenius action on the $\\ell$-torsion is a\npart of Schoof-Elkies-Atkin algorithm for point counting on an elliptic curve\n$E$ over a finite field $\\mathbb{F}_q$. The idea of Schoof's algorithm is to\ncompute the trace of Frobenius $t$ modulo primes $\\ell$ and restore it by the\nChinese remainder theorem. Atkin's improvement consists of computing the order\n$r$ of the Frobenius action on $E[\\ell]$ and of restricting the number $t\n\\pmod{\\ell}$ to enumerate by using the formula $t^2 \\equiv q (\\zeta +\n\\zeta^{-1})^2 \\pmod{\\ell}$. Here $\\zeta$ is a primitive $r$-th root of unity.\nIn this paper, we generalize Atkin's formula to the general case of abelian\nvariety of dimension $g$. Classically, finding of the order $r$ involves\nexpensive computation of modular polynomials. We study the distribution of the\nFrobenius orders in case of abelian surfaces and $q \\equiv 1 \\pmod{\\ell}$ in\norder to replace these expensive computations by probabilistic algorithms.\n

Let F F be a finite field, A = F [ T ] A=F[T] , and k = F ( T ) k=F(T) . Let K m = k ( Λ m ) K_{m}=k(\Lambda _{m}) be the field extension of k k obtained by adjoining the m m -torsion on the Carlitz module. The class number h m h_{m} of K m K_{m} can be written as a product h m = h m + h m − h_{m}=h_{m}^{+}h_{m}^{-} . The number h m − h_{m}^{-} is called the relative class number. In this paper a formula for h m − h_{m}^{-} is derived which is the analogue of the Maillet determinant formula for the relative class number of the cyclotomic field of p p -th roots of unity. Some consequences of this formula are also derived.

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].

A complete hypergeometric point count formula for Dwork hypersurfaces

In this paper, we calculate the Weiner index, some degree-based topological indices, and some eccentricity-based topological indices of Join graph of Γ(F1 ×F2 ×F3) and Γ(J1 ×J2 ×J3), where F1, F2, F3, J1, J2, J3 are finite fields of order at least two. The vertex set of Γ(F1×F2×F3) (Γ(J1×J2 ×J3)) is Z∗(F1 ×F2 ×F3) (Z∗(J1 ×J2 ×J3)), the set of non-zero zero divisors, and two distinct vertices (x1, x2, x3), (y1, y2, y3) are adjacent if and only if (x1, x2, x3) · (y1, y2, y3) = (0, 0, 0), the additive identity of the ring F1 × F2 × F3 (J1 × J2 × J3). The vertex set of Join graph Γ(F1 × F2 × F3) + Γ(J1 × J2 × J3) is V (Γ(F1 × F2 × F3)) ∪ V (Γ(J1 × J2 × J3)) and edge set is E(Γ(F1 × F2 × F3)) ∪ E(Γ(J1 × J2 × J3)) ∪ {(x1, x2, x3) · (y1, y2, y3) : (x1, x2, x3) ∈ V (Γ(F1 × F2 × F3)), (y1, y2, y3) ∈ V (Γ(J1 × J2 × J3))}.

Let Fq be the finite field of order q=ph with p>2 prime and h>1, and let Fq¯ be a subfield of Fq. From any two q¯-linearized polynomials L1,L2∈F‾q[T] of degree q, we construct an ordinary curve X(L1,L2) of genus g=(q−1)2 which is a generalized Artin–Schreier cover of the projective line P1. The automorphism group of X(L1,L2) over the algebraic closure F‾q of Fq contains a semidirect product Σ⋊Γ of an elementary abelian p-group Σ of order q2 by a cyclic group Γ of order q¯−1. We show that for L1≠L2, Σ⋊Γ is the full automorphism group Aut(X(L1,L2)) over F‾q; for L1=L2 there exists an extra involution and Aut(X(L1,L1))=Σ⋊Δ with a dihedral group Δ of order 2(q¯−1) containing Γ. Two different choices of the pair {L1,L2} may produce birationally isomorphic curves, even for L1=L2. We prove that any curve of genus (q−1)2 whose F‾q-automorphism group contains an elementary abelian subgroup of order q2 is birationally equivalent to X(L1,L2) for some separable q¯-linearized polynomials L1,L2 of degree q. We produce an analogous characterization in the special case L1=L2. This extends a result on the Artin–Mumford curves, due to Arakelian and Korchmáros [1].

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.

This thesis focuses on the interplay of extremal combinatorics and finite geometry. Combinatorics is concerned with discrete (and usually finite) objects. Extremal combinatorics studies how large or how small a collection of finite objects can be under certain restrictions. Those objects can be sets, graphs, vectors, etc. These questions are often motivated by problems in information theory and computer science. Another branch of combinatorics is finite geometry over finite fields of order q. Although there is no field of order 1, certain substructures in finite geometry can be interpreted as versions of projective spaces for q = 1. A triangle in a projective plane is a good example of this phenomenon. Following an introductory chapter, Chapter 2 gives an overview of some classical problems and their q-analogues in extremal combinatorics. The most recent results for the q-analogues of Sperner's theorem, the Erdös-Ko-Rado theorem and several versions of Bollobás's theorem are described. For the q-analogue of the Hilton-Milner theorem we give some new results. We also give a new bound for the minimum size of the q-analogue of small maximal cliques. We conclude that sometimes results for a q-analogue can be obtained by using the same technique as in the original problem. In some cases the answer to the problem is even identical to that of the original problem. In other cases techniques used for the q-analogue could be used for improving bounds for the original problem. The third chapter describes the known results for the chromatic number of the Kneser graphs and gives new bounds for the chromatic number of the q-analogue. The vertices of a Kneser graph are subsets (of a fixed size) of a set, whereas two vertices are adjacent if they are disjoint in their subset representation. In 1955 M. Kneser conjectured the chromatic number of those graphs. In 1978 this was proven correct by L. Lóvasz. Two small cases in the projective space q-analogue were solved in 2001 by J. Eisfeld, L. Storme and P. Sziklai and in 2006 by A. Chowdhury, C. Godsil and G. Royle. We describe an asymptotic result for all cases (except for one parameter family, where we give a partial proof) using the bounds from the q-analogue of the Hilton-Milner theorem. Chapters 4, 5 and 6 describe other q-analogues of the Kneser graphs. In Chapter 4 we define a family of Kneser type graphs over pairs of incident points and hyperplanes of a projective space. We describe large maximal cocliques in these graphs and prove that for small dimensions these are the largest possible cocliques. We conjecture that this is the case for all dimensions. In Chapter 5 we extend the Kneser graphs to the case of finite polar spaces instead of finite projective spaces and in some cases give descriptions and chromatic numbers of these graphs. Chapter 6 describes the generalization of Kneser graphs over coset spaces of Chevalley groups (parameterized by q) with respect to parabolic subgroups. This encompasses all previous cases and extends to some interesting new cases. First we describe these graphs when q = 1 and give chromatic numbers for some families. Then we consider the graphs defined over coset spaces of Chevalley groups for general q with respect to parabolic subgroups and study what results of the q = 1 case can be translated to the general case. In this thesis we found different possibilities for the connection between the bounds for the q = 1 case and the case for general q. In some cases the bounds for the general case are identical to the bounds for q = 1. In other cases we obtain the bounds for q = 1 by taking the limit for q¿1 in the general bound. Other cases show a completely different bound in both cases.

Commutative presemifields and semifields

Linearized rational interpolation problems at roots of unity play a crucial role in the fast and superfast Toeplitz solvers that we have developed. Our interpolation algorithm is a sequential algorithm in which a matrix polynomial that satisfies already some of the interpolation conditions is updated to satisfy two additional interpolation conditions. In the algorithm that we have used so far, the updating matrix, which is a matrix polynomial of degree one, is constructed in a two-step process that resembles Gaussian elimination. We briefly recall this approach and then consider two other approaches. The first one is a completely new approach based on an updating matrix that is unitary with respect to a discrete inner product that is based on roots of unity. The second one is an application of an algorithm for solving discrete least squares problems on the unit circle, a problem that has linearized rational interpolation at roots of unity as its limiting case. We conduct a number of numerical experiments to compare the three strategies.

In 2010, Shpilka and Volkovich established a prominent result on the equivalence of polynomial factorization and identity testing. It follows from their result that a multilinear polynomial over the finite field of order 2 can be factored in time cubic in the size of the polynomial given as a string. Later, we have rediscovered this result and provided a simple factorization algorithm based on computations over derivatives of multilinear polynomials. The algorithm has been applied to solve problems of compact representation of various combinatorial structures, including Boolean functions and relational data tables. In this paper, we describe an improvement of this factorization algorithm and report on preliminary experimental analysis.

The minimum volume of subspace trades

Let [Formula: see text] be a finite field of order [Formula: see text] and [Formula: see text] be a prime such that [Formula: see text] where [Formula: see text] are distinct odd primes with [Formula: see text] and [Formula: see text] is a positive integer with [Formula: see text]. In 2022, Rakphon et al. obtained all irreducible divisors of [Formula: see text] over [Formula: see text] where [Formula: see text] and [Formula: see text]. In this paper, we determine all primitive idempotents which can generate [Formula: see text]-constacyclic codes in [Formula: see text] where parity-check polynomials of these codes are shown in 2022. Moreover, we find the minimum Hammimg distances of the codes which are generated by primitive idempotents when [Formula: see text].