Odd Prime Power Research Articles

Ovoids of Q(6, q) of low degree

Ovoids of the parabolic quadric Q(6, q) of PG(6,q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{PG}(6,q)$$\\end{document} have been largely studied in the last 40 years. They can only occur if q is an odd prime power and there are two known families of ovoids of Q(6, q), the Thas-Kantor ovoids and the Ree-Tits ovoids, both for q a power of 3. It is well known that to any ovoid of Q(6, q) two polynomials f1(X,Y,Z)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f_1(X,Y,Z)$$\\end{document}, f2(X,Y,Z)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f_2(X,Y,Z)$$\\end{document} can be associated. In this paper we classify ovoids of Q(6, q) with max{deg(f1),deg(f2)}<(16.3q)313-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\max \\{\\deg (f_1),\\deg (f_2)\\}<(\\frac{1}{6.3}q)^{\\frac{3}{13}}-1$$\\end{document}.

Almost spanning distance trees in subsets of finite vector spaces

AbstractFor and an odd prime power , consider the vector space over the finite field , where the distance between two points and is defined as . A distance graph is a graph associated with a nonzero distance to each of its edges. We show that large subsets of vector spaces over finite fields contain every nearly spanning distance tree with bounded degree in each distance. This quantitatively improves results by Bennett, Chapman, Covert, Hart, Iosevich, and Pakianathan on finding distance paths, and results by Pham, Senger, Tait, and Thu on finding distance trees. A key ingredient in proving our main result is to obtain a colorful generalization of a classical result of Haxell about finding nearly spanning bounded‐degree trees in an expander.

Uniquely regular dessins with nilpotent automorphism groups of odd prime power order

If a graph is cut along each edge and can be divided into many faces, the graph has been defined to be a map when each face is homomorphic to an open disk and a bipartite map called dessin. A dessin D is regular Aut (D) if action transfer on the edge set. In particular, given a finite group G, a regular dessin is uniquely regular dessin if there is only one isomorphism class of Aut (D) are isomorphic to G. In this paper, for a nilpotent automorphism group of odd prime power order and nilpotency class four, we employ group-theoretical methods to classify these uniquely regular dessins.

Mutually unbiased maximally entangled bases in $$C^{d}\otimes C^{d}$$ with d an odd prime power

Mutually unbiased maximally entangled bases in $$C^{d}\otimes C^{d}$$ with d an odd prime power

The variance of a restricted sum-of-squares function over short intervals in [formula omitted

For B∈Fq[T] of degree 2n≥2, consider the number of ways of writing B=E2+γF2, where γ∈Fq⁎ is fixed, and E,F∈Fq[T] with degE=n and degF=m<n. We denote this restricted sum-of-squares function by Sγ;m(B). We obtain an exact formula for the variance of Sγ;m(B) over short intervals in Fq[T] where q is an odd prime power. Interestingly, when m+1≤n≤2m, the variance vanishes on some (not all) short intervals, which is in contrast to the standard function that counts representations as a sum of two squares. We use the method of additive characters and Hankel matrices that we previously used for the variance and correlations of the divisor function. In Section 2, we give a short overview of our approach. Section 3 gives new results relating to values of quadratic forms, over Fq, that are defined from Hankel matrices.

Laplacian eigenvalues of the unit graph of the ring [formula omitted

We investigate the Laplacian eigenvalues about unit graph G(Zn) on Zn and show the case whenever n is an even number or an odd prime power, G(Zn) would be Laplacian integral. We also prove that if n>1, then the Laplacian spectral radius of G(Zn) is equal cardinal number of V(G(Zn)) if and only if n=pk, here p is a prime integer, k is an integer that positive. In addition, this paper also characterizes n that algebraic connectivity of G(Zn) coincides with the vertex connectivity.

Two new classes of permutation trinomials over [formula omitted] with odd characteristic

Let q be an odd prime power and Fq3 be the finite field with q3 elements. In this paper, we propose two classes of permutation trinomials of Fq3 for an arbitrary odd characteristic based on the multivariate method and some subtle manipulation of solving equations with low degrees over finite fields. Moreover, we demonstrate that these two classes of permutation trinomials are QM inequivalent to all known permutation polynomials over Fq3. To the best of our knowledge, this paper is the first to study the construction of nonlinearized permutation trinomials of Fq3 with at least one coefficient lying in Fq3﹨Fq.

Cycles of quadratic Latin squares and antiperfect 1‐factorisations

AbstractA Latin square of order is an matrix of symbols, such that each symbol occurs exactly once in each row and column. For an odd prime power let denote the finite field of order . A quadratic Latin square is a Latin square defined by for some such that and are quadratic residues in . Quadratic Latin squares have previously been used to construct perfect 1‐factorisations, mutually orthogonal Latin squares and atomic Latin squares. We first characterise quadratic Latin squares which are devoid of Latin subsquares. Let be a graph and a 1‐factorisation of . If the union of every pair of 1‐factors in induces a Hamiltonian cycle in then is called perfect, and if there is no pair of 1‐factors in which induce a Hamiltonian cycle in then is called antiperfect. We use quadratic Latin squares to construct new examples of antiperfect 1‐factorisations of complete graphs and complete bipartite graphs. We also demonstrate that for each odd prime , there are only finitely many orders , which are powers of , such that quadratic Latin squares of order could be used to construct perfect 1‐factorisations of complete graphs or complete bipartite graphs.

A note on complex conference matrices

Let Fq be the Galois field of order q=pm, p a prime number and m a positive integer. We prove in this article that for any nontrivial multiplicative character ϰ of Fq⁎ and for any b∈Fq⁎ we have∑a∈Fq⁎ϰ(a)ϰ(a+b)‾=−1. Whenever q is odd and ϰ is the Legendre symbol this formula reduces to the well-known Jacobsthal's formula. A complex conference matrix is a square matrix of order n with zero diagonal and unimodular complex numbers elsewhere such that C⁎C=(n−1)I. Paley used finite fields with odd orders q=pm, p prime and the real Legendre symbol to construct real symmetric conference matrices of orders q+1 whenever q≡1(mod4) and real skew-symmetric conference matrices of orders q+1 whenever q≡−1(mod4). In this article we extend Paley construction to the complex setting. We extend Jacobsthal's formula to all other nontrivial characters to produce a complex symmetric conference matrix of order q+1 whenever q≥4 is any prime power as well as a complex skew-symmetric conference matrix of order q+1 whenever q is any odd prime power. These matrices were constructed very recently in connection with harmonic Grassmannian codes, by use of finite fields and the character table of their additive characters. We propose here a new proof of their construction by use of the above generalized formula similarly as was done by Paley in the real case. We also classify, up to equivalence, the complex conference matrices constructed with some nontrivial characters. In particular, we prove that the complex conference matrix constructed with any nontrivial multiplicative character ϰ and that one constructed with ϰpk for any integer k=1,...m−1 are permutation equivalent. Moreover, we determine the spectrum of any complex conference matrix obtained from this construction.

Block-Graceful Designs

In this article, we adapt the edge-graceful graph labeling definition into block designs and define a block design V , B with V = v and B = b as block-graceful if there exists a bijection f : B ⟶ 1,2 , … , b such that the induced mapping f + : V ⟶ Z v given by f + x = ∑ x ∈ A A ∈ B f A mod v is a bijection. A quick observation shows that every v , b , r , k , λ − BIBD that is generated from a cyclic difference family is block-graceful when v , r = 1 . As immediate consequences of this observation, we can obtain block-graceful Steiner triple system of order v for all v ≡ 1 mod 6 and block-graceful projective geometries, i.e., q d + 1 − 1 / q − 1 , q d − 1 / q − 1 , q d − 1 − 1 / q − 1 − BIBDs. In the article, we give a necessary condition and prove some basic results on the existence of block-graceful v , k , λ − BIBDs. We consider the case v ≡ 3 mod 6 for Steiner triple systems and give a recursive construction for obtaining block-graceful triple systems from those of smaller order which allows us to get infinite families of block-graceful Steiner triple systems of order v for v ≡ 3 mod 6 . We also consider affine geometries and prove that for every integer d ≥ 2 and q ≥ 3 , where q is an odd prime power or q = 4 , there exists a block-graceful q d , q , 1 − BIBD. We make a list of small parameters such that the existence problem of block-graceful labelings is completely solved for all pairwise nonisomorphic BIBDs with these parameters. We complete the article with some open problems and conjectures.

Packing cycles in undirected group-labelled graphs

We prove a refinement of the flat wall theorem of Robertson and Seymour to undirected group-labelled graphs (G,γ) where γ assigns to each edge of an undirected graph G an element of an abelian group Γ. As a consequence, we prove that Γ-nonzero cycles (cycles whose edge labels sum to a non-identity element of Γ) satisfy the half-integral Erdős-Pósa property, and we also recover a result of Wollan that if Γ has no element of order two, then Γ-nonzero cycles satisfy the Erdős-Pósa property. As another application, we prove that if m is an odd prime power, then cycles of length ℓmodm satisfy the Erdős-Pósa property for all integers ℓ. This partially answers a question of Dejter and Neumann-Lara from 1987 on characterizing all such integer pairs (ℓ,m).

ON THE NUMBER OF QUADRATIC ORTHOMORPHISMS THAT PRODUCE MAXIMALLY NONASSOCIATIVE QUASIGROUPS

AbstractLet q be an odd prime power and suppose that $a,b\in \mathbb {F}_q$ are such that $ab$ and $(1{-}a)(1{-}b)$ are nonzero squares. Let $Q_{a,b} = (\mathbb {F}_q,*)$ be the quasigroup in which the operation is defined by $u*v=u+a(v{-}u)$ if $v-u$ is a square, and $u*v=u+b(v{-}u)$ if $v-u$ is a nonsquare. This quasigroup is called maximally nonassociative if it satisfies $x*(y*z) = (x*y)*z \Leftrightarrow x=y=z$ . Denote by $\sigma (q)$ the number of $(a,b)$ for which $Q_{a,b}$ is maximally nonassociative. We show that there exist constants $\alpha \approx 0.029\,08$ and $\beta \approx 0.012\,59$ such that if $q\equiv 1 \bmod 4$ , then $\lim \sigma (q)/q^2 = \alpha $ , and if $q \equiv 3 \bmod 4$ , then $\lim \sigma (q)/q^2 = \beta $ .

Diameter, edge-connectivity, and C4-freeness

Improving a recent result of Fundikwa, Mazorodze, and Mukwembi, we show that d≤(2n−3)/5 for every connected C4-free graph of order n, diameter d, and edge-connectivity at least 3, which is best possible up to a small additive constant. For edge-connectivity at least 4, we improve this to d≤(n−1)/3. Furthermore, adapting a construction due to Erdős, Pach, Pollack, and Tuza, for an odd prime power q at least 7, and every positive integer k, we show the existence of a connected C4-free graph of order n=(q2+q−1)k+1, diameter d=4k, and edge-connectivity λ at least q−6, that is, d≥4(n−1)/(λ2+O(λ)) for the constructed graph.

Construction of quasi-cyclic self-dual codes over finite fields

ABSTRACT Our goal of this paper is to find a construction of all ℓ-quasi-cyclic self-dual codes over a finite field F q of length mℓ for every positive even integer ℓ. In this paper, we study the case where x m − 1 has an arbitrary number of irreducible factors in F q [ x ] ; in the previous studies, only some special cases where x m − 1 has exactly two or three irreducible factors in F q [ x ] , were studied. Firstly, the binary code case is completed: for any even positive integer ℓ, every binary ℓ-quasi-cyclic self-dual code can be obtained by our construction. Secondly, we work on the q-ary code cases for an odd prime power q. We find an explicit method for construction of all ℓ-quasi-cyclic self-dual codes over F q of length mℓ for any even positive integer ℓ, where we require that q ≡ 1 ( mod 4 ) if the index ℓ ≥ 6 . By implementation of our method, we obtain a new optimal binary self-dual code [ 172 , 86 , 24 ] , which is also a quasi-cyclic code of index 4.

On a family of linear MRD codes with parameters $$[8\times 8,16,7]_q$$

In this paper we consider a family \(\mathcal {F}\) of 2n-dimensional \({\mathbb {F}}_q\)-linear rank metric codes in \({\mathbb {F}}_q^{n\times n}\) arising from polynomials of the form \(x^{q^s}+\delta x^{q^{\frac{n}{2}+s}}\in {\mathbb {F}}_{q^n}[x]\). The family \(\mathcal {F}\) was introduced by Csajbók et al. (JAMA 548:203–220) as a potential source for maximum rank distance (MRD) codes. Indeed, they showed that \(\mathcal {F}\) contains MRD codes for \(n=8\), and other subsequent partial results have been provided in the literature towards the classification of MRD codes in \(\mathcal {F}\) for any n. In particular, the classification has been reached when n is smaller than 8, and also for n greater than 8 provided that s is small enough with respect to n. In this paper we deal with the open case \(n=8\), providing a classification for any large enough odd prime power q. The techniques are from algebraic geometry over finite fields, since our strategy requires the analysis of certain 3-dimensional \({\mathbb {F}}_q\)-rational algebraic varieties in a 7-dimensional projective space. We also show that the MRD codes in \(\mathcal {F}\) are not equivalent to any other MRD codes known so far.

Obstructions to reciprocity laws related to division fields of Drinfeld modules

Let q q be an odd prime power, denote by F q \mathbb {F}_q the finite field with q q elements, and set A ≔ F q [ T ] A ≔\mathbb {F}_q[T] , F ≔ F q ( T ) F ≔\mathbb {F}_q(T) . Let ψ : A → F { τ } \psi : A \to F\{\tau \} be a Drinfeld A A -module over F F , of rank r ≥ 2 r \geq 2 , with E n d F ¯ ( ψ ) = A End_{\overline {F}}(\psi ) = A . For a non-zero ideal n \mathfrak {n} of A A , denote its unique monic generator by n n , denote the degree of n n as a polynomial in T T by deg ⁡ n \deg n , and denote the n \mathfrak {n} -division field of ψ \psi by F ( ψ [ n ] ) F(\psi [\mathfrak {n}]) . A reciprocity law for ψ \psi asserts that, if gcd ( c h a r F , r ) = 1 \gcd (charF, r) = 1 or if n \mathfrak {n} is prime, then a non-zero prime ideal p ∤ n \mathfrak {p} \nmid \mathfrak {n} of A A splits completely in F ( ψ [ n ] ) F(\psi [\mathfrak {n}]) if and only if the Frobenius trace a 1 , p ( ψ ) a_{1, \mathfrak {p}}(\psi ) of ψ \psi at p \mathfrak {p} and the first component b 1 , p ( ψ ) b_{1, \mathfrak {p}}(\psi ) of the Frobenius index of ψ \psi at p \mathfrak {p} satisfy the congruences a 1 , p ( ψ ) ≡ − r ( mod n ) a_{1, \mathfrak {p}}(\psi ) \equiv -r \pmod n and b 1 , p ( ψ ) ≡ 0 ( mod n ) b_{1, \mathfrak {p}}(\psi ) \equiv 0 \pmod n . We find the Dirichlet density of the set of non-zero prime ideals p \mathfrak {p} for which the latter congruence never holds, that is, for which b 1 , p ( ψ ) = 1 b_{1, \mathfrak {p}}(\psi ) = 1 . Using similar methods, we prove an asymptotic formula for the function of x x defined by the average 1 # { p : deg ⁡ p = x } ∑ p : deg ⁡ p = x τ A ( b 1 , p ( ψ ) ) \frac {1}{\#\{\mathfrak {p}: \ \deg p = x \}} \sum _{\mathfrak {p}: \ \deg p = x} \tau _A(b_{1, \mathfrak {p}}(\psi )) , where p = A p \mathfrak {p} = A p denotes an arbitrary non-zero prime ideal of A A whose monic generator p ∈ A p \in A has degree x x and where τ A ( b 1 , p ( ψ ) ) \tau _A(b_{1, \mathfrak {p}}(\psi )) denotes the number of monic divisors of b 1 , p ( ψ ) b_{1, \mathfrak {p}}(\psi ) .

Optimal strong approximation for quadrics over [formula omitted

Suppose q is a fixed odd prime power, F(x) is a non-degenerate quadratic form over Fq[t] of discriminant Δ in d≥5 variables x, and f,g∈Fq[t], λ∈Fq[t]d. We show that whenever deg⁡f≥(4+ε)deg⁡g+Oε,F(1), gcd⁡(Δ∞,fg)=O(1), and the necessary local conditions are satisfied, we have a solution x∈Fq[t]d to F(x)=f such that x≡λmodg. For d=4, we show that the same conclusion holds if we instead have deg⁡f≥(6+ε)deg⁡g+Oε,F(1). This gives us a new proof (independent of the Ramanujan conjecture over function fields proved by Drinfeld) that the diameter of any k-regular Morgenstern Ramanujan graph G is at most (2+ε)logk−1⁡|G|+Oε(1). In contrast to the d=4 case, our result is optimal for d≥5. Our main new contributions are a stationary phase theorem over function fields for bounding oscillatory integrals, and a notion of anisotropic cones to circumvent isotropic phenomena in the function field setting.

A characterization of the family of secant lines to a hyperbolic quadric in PG(3,q), q odd, Part II

In [9], two of us classified line sets in PG(3,q), q odd, that satisfy a certain list of properties. It was shown there that if q≥7, then each such line set is either the set of secant lines with respect to a hyperbolic quadric of PG(3,q) or belongs to a certain “hypothetical family” of line sets (for which no examples were known in [9]). In the present paper, we achieve two goals. On the one hand, we extend the mentioned classification result to all odd prime powers q. On the other hand, we study the hypothetical family of line sets and show that they are related to quadratic sets of the Klein quadric. This will allow us to show that such line sets exist for every odd prime power q.

On the automorphisms of a family of small q-regular graphs of girth 8

In this paper we investigate the automorphisms of a family of small (q,8)-graphs of order 2q3 − 2q which are obtained as induced subgraphs of the incidence graph of the classical generalized quadrangle of order q. We show that for q an odd prime power, the automorphism group has four orbits on the set of vertices, thus the investigated graphs cannot be Cayley graphs or lifts of a dipole.

Arithmetic correlation of binary half‐ℓ‐sequences

AbstractThe arithmetic correlations of two binary half‐ℓ‐sequences with connection integer pr, which is an odd prime power, are investigated. Possible values (of the arithmetic correlation) are calculated. In particular, if p ≡ 1 (mod 8), the authors prove that they are zero for non‐trivial shifts, that is, the half‐ℓ‐sequences have ideal arithmetic correlations. If p ≡ −1 (mod 8), an upper bound, which is of order of magnitude pr−1/2 ln p, is derived by using earlier results on the imbalance of half‐ℓ‐sequences with connection integer p studied by Gu and Klapper and later improved by Wang and Tan.