Odd Prime Power Research Articles

Isomorphisms of generalized Cayley graphs

In this paper, we investigate the isomorphism problems of the generalized Cayley graphs, which are generalizations of the traditional Cayley graphs. We find that there are two types of natural isomorphisms for the generalized Cayley graphs. We also study the GCI-groups among the generalized Cayley graphs, and the Cayley regressions of some groups. We mainly showed that, for an odd prime power n, Z2n (resp. D2n) is a restricted GCI-group if D2n (resp. Z2n) is a CI-group. We also obtain that the cyclic group of order 2n is a 4-quasi-Cayley regression if and only if n = 3.

Almost difference sets in nonabelian groups

We give two new constructions of almost difference sets. The first is a generic construction of $$(q^{2}(q+1),\,q(q^{2}-1),\,q(q^{2}-q-1),\,q^{2}-1)$$ almost difference sets in certain groups of order $$q^{2}(q+1)$$ (q is an odd prime power) having ( $$\mathbb {F}_{q^{2}},\,+)$$ as a subgroup. This construction yields several infinite families of almost difference sets, many of which occur in nonabelian groups. The second construction yields $$(4p,\,2p+1,\,p,\,p-1)$$ almost difference sets in dihedral groups of order 4p where $$p\equiv 3 (\mathrm{mod} 4)$$ is a prime. Moreover, it turns out that some of the infinite families of almost difference sets obtained produce Cayley graphs which are Ramanujan graphs.

On eigenfunctions and maximal cliques of Paley graphs of square order

In this paper we present a family of maximal cliques of size q+12 or q+32, accordingly as q≡1(4) or q≡3(4), in Paley graphs of order q2, where q is an odd prime power. After that we use the new cliques to define a family of eigenfunctions corresponding to both non-principal eigenvalues and having support size q+1, which is the minimum possible value by the weight-distribution bound. Finally, we prove that the constructed eigenfunction comes from an equitable partition.

ON LINEAR COMPLEMENTARY DUAL FOUR CIRCULANT CODES

We study linear complementary dual four circulant codes of length $4n$ over $\mathbb{F}_{q}$ when $q$ is an odd prime power. When $q^{\unicode[STIX]{x1D6FF}}+1$ is divisible by $n$, we obtain an exact count of linear complementary dual four circulant codes of length $4n$ over $\mathbb{F}_{q}$. For certain values of $n$ and $q$ and assuming Artin’s conjecture for primitive roots, we show that the relative distance of these codes satisfies a modified Gilbert–Varshamov bound.

Constacyclic codes over the ring $${\mathbb {F}}_q+v{\mathbb {F}}_q+v^{2}{\mathbb {F}}_q$$ F q + v F q + v 2 F q and their applications of constructing new non-binary quantum codes

Let $$R={\mathbb {F}}_q+v{\mathbb {F}}_q+v^{2}{\mathbb {F}}_q$$ be a finite non-chain ring, where q is an odd prime power and $$v^3=v$$ . In this paper, we propose two methods of constructing quantum codes from $$(\alpha +\beta v+\gamma v^{2})$$ -constacyclic codes over R. The first one is obtained via the Gray map and the Calderbank–Shor–Steane construction from Euclidean dual-containing $$(\alpha +\beta v+\gamma v^{2})$$ -constacyclic codes over R. The second one is obtained via the Gray map and the Hermitian construction from Hermitian dual-containing $$(\alpha +\beta v+\gamma v^{2})$$ -constacyclic codes over R. As an application, some new non-binary quantum codes are obtained.

A new efficient MEP.1 series for [formula omitted] factorial experiments

Main effect plus one plans (MEP.1) search for a non-zero interaction effect and estimate it in addition to estimate all main effects. A new series of MEP.1 with 2f runs has been constructed and given in this research for 2f factorial experiments for all odd prime power f≥9. Designs in this series are more efficient in estimating the factorial effects than the previous existent MEP.1 with the same number of runs based on D-efficiency criterion. In contrast to the former designs with 2f runs, the D-efficiency of new proposed designs is increasing in f.

Irreducible compositions of degree two polynomials over finite fields have regular structure

Let $q$ be an odd prime power and $D$ be the set of monic irreducible polynomials in $\mathbb F_q[x]$ which can be written as a composition of monic degree two polynomials. In this paper we prove that $D$ has a natural regular structure by showing that there exists a finite automaton having $D$ as accepted language. Our method is constructive.

Characterization of intersecting families of maximum size in PSL(2,q)

We consider the action of the 2-dimensional projective special linear group PSL(2,q) on the projective line PG(1,q) over the finite field Fq, where q is an odd prime power. A subset S of PSL(2,q) is said to be an intersecting family if for any g1,g2∈S, there exists an element x∈PG(1,q) such that xg1=xg2. It is known that the maximum size of an intersecting family in PSL(2,q) is q(q−1)/2. We prove that all intersecting families of maximum size are cosets of point stabilizers for all odd prime powers q>3.

Incidence Matrices of Finite Quadratic Spaces

Let q be an odd prime power, and PG(n,q) be the projective space of ${\mathbb {F}}_{q}^{n + 1}$ . We equip ${\mathbb {F}}_{q}^{n + 1}$ with a nondegenerate quadratic form. The 2-rank of the incidence matrix of anisotropic points versus their corresponding hyperplanes of PG(n,q) is determined for n = 2,3.

Powerful numbers in (1ℓ + qℓ)(2ℓ + qℓ)⋯(nℓ + qℓ)

Let q be a positive integer. Recently, Niu and Liu proved that, if n≥max⁡{q,1198−q}, then the product (13+q3)(23+q3)⋯(n3+q3) is not a powerful number. In this note, we prove (1) that, for any odd prime power ℓ and n≥max⁡{q,11−q}, the product (1ℓ+qℓ)(2ℓ+qℓ)⋯(nℓ+qℓ) is not a powerful number, and (2) that, for any positive odd integer ℓ, there exists an integer Nq,ℓ such that, for any positive integer n≥Nq,ℓ, the product (1ℓ+qℓ)(2ℓ+qℓ)⋯(nℓ+qℓ) is not a powerful number.

On LCD Negacyclic Codes over Finite Fields

In this paper, by investigating the factor of the $x^n+1$, we deduce that the structure of the reversible negacyclic code over the finite field $\mathbb{F}_{q}$, where $q$ is an odd prime power. Though studying $q-$cyclotomic cosets modulo $2n$, we obtain the parameters of negacyclic BCH code of length $n=\frac{q^\ell+1}{2}$ , $n=\frac{q^m-1}{2(q-1)}$ and $n=\frac{q^{t\cdot2^\tau}-1}{2(q^t+1)}$. Some optimal linear codes from negacyclic codes are given. Finally, we discuss a class of MDS LCD negacyclic codes.

A class of negacyclic BCH codes and its application to quantum codes

In this paper, we study negacyclic BCH codes over \(\mathbb {F}_{q}\) of length \(n=(q^{2m}-1)/(q-1)\), where q is an odd prime power and m is a positive integer. In particular, the dimension, the minimum distance and the weight distribution of some negacyclic BCH codes over \(\mathbb {F}_{q}\) of length \(n=(q^{2m}-1)/(q-1)\) are determined. Two classes of negacyclic BCH codes meeting the Griesmer bound are obtained. As an application, we construct quantum codes with good parameters from this class of negacyclic BCH codes.

Weak metacirculants of odd prime power order

Metacirculants are a basic and well-studied family of vertex-transitive graphs, and weak metacirculants are generalizations of them. A graph is called a weak metacirculant if it has a vertex-transitive metacyclic automorphism group. This paper is devoted to the study of weak metacirculants with odd prime power order. We first prove that a weak metacirculant of odd prime power order is a metacirculant if and only if it has a vertex-transitive split metacyclic automorphism group. We then prove that for any odd prime p and integer ℓ≥4, there exist weak metacirculants of order pℓ which are Cayley graphs but not Cayley graphs of any metacyclic group; this answers a question in Li et al. (2013) [11] We construct such graphs explicitly by introducing a construction which is a generalization of generalized Petersen graphs. Finally, we determine all smallest possible metacirculants of odd prime power order which are Cayley graphs but not Cayley graphs of any metacyclic group.

Structure of group invariant weighing matrices of small weight

We show that every weighing matrix of weight n invariant under a finite abelian group G can be generated from a subgroup H of G with |H|≤2n−1. Furthermore, if n is an odd prime power and a proper circulant weighing matrix of weight n and order v exists, then v≤2n−1. We also obtain a lower bound on the weight of group invariant matrices depending on the invariant factors of the underlying group. These results are obtained by investigating the structure of subsets of finite abelian groups that do not have unique differences.

A note on commutative semifield planes

Abstract Let q be an odd prime power. We prove that a planar function f from 𝔽 q to itself can be written as an affine Dembowski–Ostrom polynomial if and only if the projective plane derived from f is a commutative semifield plane.

Steiner’s Porism in finite Miquelian Möbius planes

Abstract We investigate Steiner’s Porism in finite Miquelian Möbius planes constructed over the pair of finite fields GF(q) and GF(q 2), for an odd prime power q. Properties of common tangent circles for two given concentric circles are discussed and with that, a finite version of Steiner’s Porism for concentric circles is stated and proved. We formulate conditions on the length of a Steiner chain by using the quadratic residue theorem in GF(q). These results are then generalized to an arbitrary pair of non-intersecting circles by introducing the notion of capacitance, which turns out to be invariant under Möbius transformations. Finally, the results are compared with the situation in the classical Euclidean plane.

Circulant Homogeneous Factorisations of Complete Digraphs $\rm\bf K_{p^d}$ with $p$ an Odd Prime

Let $\mathcal F=(\rm\bf K_{n},\mathcal P)$ be a circulant homogeneous factorisation of index $k$, that means $\mathcal P$ is a partition of the arc set of the complete digraph $\rm\bf K_n$ into $k$ circulant factor digraphs such that there exists $\sigma\in S_n$ permuting the factor circulants transitively amongst themselves. Suppose further such an element $\sigma$ normalises the cyclic regular automorphism group of these circulant factor digraphs, we say $\mathcal F$ is normal. Let $\mathcal F=(\rm\bf K_{p^d},\mathcal P)$ be a circulant homogeneous factorisation of index $k$ where $p^d$, ($d\ge 1$) is an odd prime power. It is shown in this paper that either $\mathcal F$ is normal or $\mathcal F$ is a lexicographic product of two smaller circulant homogeneous factorisations.

Linear codes from quadratic forms

Linear codes have been an interesting topic in both theory and practice for many years. In this paper, for an odd prime power q, we present a class of linear codes over finite fields \(F_q\) with quadratic forms via a general construction and then determine the explicit complete weight enumerators of these linear codes. Our construction covers some related ones via quadratic form functions and the linear codes may have applications in cryptography and secret sharing schemes.

On the locality of arb-invariant first-order formulas with modulo counting quantifiers

We study Gaifman locality and Hanf locality of an extension of first-order logic with modulo p counting quantifiers (FO+MOD_p, for short) with arbitrary numerical predicates. We require that the validity of formulas is independent of the particular interpretation of the numerical predicates and refer to such formulas as arb-invariant formulas. This paper gives a detailed picture of locality and non-locality properties of arb-invariant FO+MOD_p. For example, on the class of all finite structures, for any p >= 2, arb-invariant FO+MOD_p is neither Hanf nor Gaifman local with respect to a sublinear locality radius. However, in case that p is an odd prime power, it is weakly Gaifman local with a polylogarithmic locality radius. And when restricting attention to the class of string structures, for odd prime powers p, arb-invariant FO+MOD_p is both Hanf and Gaifman local with a polylogarithmic locality radius. Our negative results build on examples of order-invariant FO+MOD_p formulas presented in Niemist\"o's PhD thesis. Our positive results make use of the close connection between FO+MOD_p and Boolean circuits built from NOT-gates and AND-, OR-, and MOD_p- gates of arbitrary fan-in.

A congruence involving harmonic sums modulo pαqβ

In 2014, Wang and Cai established the following harmonic congruence for any odd prime [Formula: see text] and positive integer [Formula: see text], [Formula: see text] where [Formula: see text] and [Formula: see text] denote the set of positive integers which are prime to [Formula: see text]. In this paper, we obtain an unexpected congruence for distinct odd primes [Formula: see text], [Formula: see text] and positive integers [Formula: see text], [Formula: see text] and the necessary and sufficient condition for [Formula: see text] Finally, we raise a conjecture that for [Formula: see text] and odd prime power [Formula: see text], [Formula: see text], [Formula: see text] However, we fail to prove it even for [Formula: see text] with three distinct prime factors.