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

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].

Let Fq be a finite field of order q and t be a prime such that q t − 1 = l v1 1 l v2 2 l v3 3 c where l1, l2, l3 are distinct odd primes with gcd(l1l2l3, q(q−1)) = 1 and c is a positive integer with gcd(l1l2l3, c) = 1. In this thesis, we find all irreducible divisors of x l m1 1 l m2 2 l m3 3 −a over Fq and all generator polynomials for constructing a-constacyclic codes of length l m1 1 l m2 2 l m3 3 . In addition, we determine all primitive idempotents for irreducible constacyclic codes of length l m1 1 l m2 2 l m3 3 and find the minimum Hammimg distances of the codes which are generated by primitive idempotents when t = 2.

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].

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

The fundamental theorem of algebra states that every non-constant polynomial with complex coefficients can be factorized on linear factors with coefficients from the same field. However, this is not true for numerous other equally important and interesting fields such as, for example, finite fields. All existing finite fields are of the q=pk order, where p is a prime number and k is a positive integer. Any two finite fields of the same order are mutually isomorfic, therefore the field of q order is unique; it is also known as »the Galois field«, named after the French mathematician Evariste Galois (1811-1832). This thesis deals with factorization of polynomials over finite fields. The general rule states that some polynomials are not reducible, yet each polynomial with coefficients of the field can be uniquely factorized into irreducible factors. There are various known algorithms that can determine whether or not a polynomial is reducible and – in the case of reducible polynomials – also find this factorization. There are several such algorithms, however, this thesis shall focus on the Berlekamp's algorithm. The introductory chapter introduces the historical development of the various approaches to polynomial factorization and presents some modern applications that are currently in use. The second chapter focuses on basic concepts and characteristics of algebraic structures and some classic results that are crucial for the understanding of the algorithm (e.g. groups, rings, fields, division rings and polynomial rings.). The algorithm also uses the Chinese remainder theorem for polynomials and Euclidean algorithm for calculating the greatest common divisor of two polynomials. The first part of Chapter 3 (i.e. the main chapter of the thesis) demonstrates 4 naive methods for the finding of polynomial factorization. For each of the methods we first introduced factorization of a polynomial over real numbers and then continued with an example of a polynomial with coefficients in the finite field. The sieve method, for instance, builds a list of irreducible polynomials of low degrees. Irreducible polynomials up to (and including) degree 6 over the finite fields Z2, Z3, Z5 and Z7 can be calculated using our own code made in the VisualC# programming language. These naive methods are followed by a detailed description of the Berlekamp's algorithm with theorems, proofs and various examples. At the end of section we present our implementation of the algorithm in the computational algebra system MAGMA.

Let 𝔽q stand for the finite field of odd characteristic p with q elements (q = pn, n ∈ ℕ) and [Formula: see text] denote the set of all the nonzero elements of 𝔽q. Let m and t be positive integers. By using the Smith normal form of the exponent matrix, we obtain a formula for the number of rational points on the variety defined by the following system of equations over [Formula: see text] where the integers t > 0, r0 = 0 < r1 < r2 < ⋯ < rt, 1 ≤ n1 < n2 <, ⋯ < nt and 0 ≤ j ≤ t − 1, bk ∊ 𝔽q, ak,i ∊ [Formula: see text] (k = 1, …, m, i = 1, …, rt), and the exponent of each variable is a positive integer. Further, under some natural conditions, we arrive at an explicit formula for the number of 𝔽q-rational points on the above variety. It extends the results obtained previously by Wolfmann, Sun, Wang, Hong et al. Our result gives a partial answer to an open problem raised in [The number of rational points of a family of hypersurfaces over finite fields, J. Number Theory 156 (2015) 135–153].

Autocorrelation Function of a Family of Complex Sequences

Let G be a simple connected graph with r vertices and t edges. Let d be a positive integer and \(a\in F_{p^n}^{*}\). The associated system of equations S(G, d, p n , a) of G over a finite field \(F_{p^n}^{*}\) is defined to be AX = C, where \(X = (x^d_1, x^d_2,...,x^d_t)^{\top}\), C = (a, a, ..., a) ⊤ , A = (a ij ) is the incidence matrix of G. We show that the solutions of S(G, d, p n , a) has a relation with the number of perfect matchings of G. We also show that the problem of determining the number of solutions to a system of equations over finite field is NP-hard.

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

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