Odd Prime Power Research Articles

Formally dual subsets of cyclic groups of prime power order

We study the notion of formal-duality over finite cyclic groups of prime power order as introduced by Cohn, Kumar, Reiher and Sch\"urmann. We will prove that for any cyclic group of odd prime power order, as well as for any cyclic group of order $2^{2l+1}$, there is no primitive pair of formally-dual subsets. This partially proves a conjecture, made by the priorly mentioned authors, that the only cyclic groups with a pair of primitive formally-dual subsets are $\{0\}$ and $\mathbb{Z}/4\mathbb{Z}$.

New nonbinary quantum codes with larger distance constructed from BCH codes over 𝔽q2

This paper concentrates on construction of new nonbinary quantum error-correcting codes (QECCs) from three classes of narrow-sense imprimitive BCH codes over finite field [Formula: see text] ([Formula: see text] is an odd prime power). By a careful analysis on properties of cyclotomic cosets in defining set [Formula: see text] of these BCH codes, the improved maximal designed distance of these narrow-sense imprimitive Hermitian dual-containing BCH codes is determined to be much larger than the result given according to Aly et al. [S. A. Aly, A. Klappenecker and P. K. Sarvepalli, IEEE Trans. Inf. Theory 53, 1183 (2007)] for each different code length. Thus families of new nonbinary QECCs are constructed, and the newly obtained QECCs have larger distance than those in previous literature.

A class of constacyclic BCH codes and new quantum codes

Constacyclic BCH codes have been widely studied in the literature and have been used to construct quantum codes in latest years. However, for the class of quantum codes of length $$n=q^{2m}+1$$n=q2m+1 over $$F_{q^2}$$Fq2 with q an odd prime power, there are only the ones of distance $$\delta \le 2q^2$$?≤2q2 are obtained in the literature. In this paper, by a detailed analysis of properties of $$q^{2}$$q2-ary cyclotomic cosets, maximum designed distance $$\delta _\mathrm{{max}}$$?max of a class of Hermitian dual-containing constacyclic BCH codes with length $$n=q^{2m}+1$$n=q2m+1 are determined, this class of constacyclic codes has some characteristic analog to that of primitive BCH codes over $$F_{q^2}$$Fq2. Then we can obtain a sequence of dual-containing constacyclic codes of designed distances $$2q^2 2q^2$$d>2q2 can be constructed from these dual-containing codes via Hermitian Construction. These newly obtained quantum codes have better code rate compared with those constructed from primitive BCH codes.

Improved Constructions for Nonbinary Quantum BCH Codes

In this work, we present two improved constructions for nonbinary quantum BCH codes of lengths \(n=\frac {q^{4}-1}{2}\) and \(n=\frac {q^{4}-1}{q-1}\), where q is an odd prime power. Moreover, the constructed quantum BCH codes have parameters better than those obtained from other known constructions.

A New Infinite Family of Hemisystems of the Hermitian Surface

In this paper, we construct an infinite family of hemisystems of the Hermitian surface $\mathsf{H}(3,q^2)$. In particular, we show that for every odd prime power $q$ congruent to $3$ modulo $4$, there exists a hemisystem of $\mathsf{H}(3,q^2)$ admitting $C_{(q^3+1)/4} : C_3$.

Polarity graphs and Ramsey numbers for [formula omitted] versus stars

For two given graphs G1 and G2, the Ramsey number R(G1,G2) is the smallest integer N such that for any graph of order N, either G contains a copy of G1 or its complement contains a copy of G2. Let Cm be a cycle of length m and K1,n a star of order n+1. Parsons (1975) shows that R(C4,K1,n)≤n+⌊n−1⌋+2 and if n is the square of a prime power, then the equality holds. In this paper, by discussing the properties of polarity graphs whose vertices are points in the projective planes over Galois fields, we prove that R(C4,K1,q2−t)=q2+q−(t−1) if q is an odd prime power, 1≤t≤2⌈q4⌉ and t≠2⌈q4⌉−1, which extends a result on R(C4,K1,q2−t) obtained by Parsons (1976).

A family of mixed graphs with large order and diameter 2

A mixed regular graph is a connected simple graph in which each vertex has both a fixed outdegree (the same indegree) and a fixed undirected degree. A mixed regular graphs is said to be optimal if there is not a mixed regular graph with the same parameters and bigger order.We present a construction that provides mixed graphs of undirected degree q, directed degree q−12 and order 2q2, for q being an odd prime power. Since the Moore bound for a mixed graph with these parameters is equal to 9q2−4q+34 the defect of these mixed graphs is (q−22)2−14.In particular we obtain a known mixed Moore graph of order 18, undirected degree 3 and directed degree 1 called Bosák’s graph and a new mixed graph of order 50, undirected degree 5 and directed degree 2, which is proved to be optimal.

The Existence of Minimal Logarithmic Signatures for Some Finite Simple Groups

A logarithmic signature for a finite group G is a sequence α = [A1, …, As] of subsets of G such that every element g ∈ G can be uniquely written in the form g = g1…gs, where gi ∈ Ai, 1 ⩽ i ⩽ s. The number ∑si = 1|Ai| is called the length of α and denoted by l(α). A logarithmic signature α is said to be minimal (MLS) if l(α) = ∑ni = 1mipi, where is the prime factorization of |G|. The MLS conjecture states that every finite simple group has an MLS. The aim of this article is proving the existence of a minimal logarithmic signature for the untwisted groups G2(3n), the orthogonal groups Ω7(q) and PΩ+8(q), q is an odd prime power, the orthogonal groups Ω9(3), PΩ+10(3), and PΩ−8(3), the Tits simple group 2F4(2)′, the Janko group J3, the twisted group 3D4(2), the Rudvalis group Ru, and the Fischer group Fi22. As a consequence of our results, it is proved that all finite groups of order ⩽ 1012 other than the Ree group Ree(27), the O’Nan group O′N, and the untwisted group G2(7) have MLS.

A construction of dense mixed graphs of diameter 2

A mixed graph is said to be dense, if its order is close to the Moore bound and it is optimal if there is not a mixed graph with the same parameters and bigger order. We give a construction that provides dense mixed graphs of undirected degree q, directed degree q−12 and order 2q2, for q being an odd prime power. Since the Moore bound for a mixed graph with these parameters is equal to 9q2−4q+34 the defect of these mixed graphs is (q−22)2−14. In particular we obtain a known mixed Moore graph of order 18, undirected degree 3 and directed degree 1, called Bosák's graph and a new mixed graph of order 50, undirected degree 5 and directed degree 2, which is proved to be optimal.

Restriction of Odd Degree Characters and Natural Correspondences

Let |$q$| be an odd prime power, |$n > 1$|⁠, and let |$P$| denote a maximal parabolic subgroup of |$GL_n(q)$| with Levi subgroup |$GL_{n-1}(q) \times GL_1(q)$|⁠. We restrict the odd-degree irreducible characters of |$GL_n(q)$| to |$P$| to discover a natural correspondence of characters, both for |$GL_n(q)$| and |$SL_n(q)$|⁠. A similar result is established for certain finite groups with self-normalizing Sylow |$p$|-subgroups. Next, we construct a canonical bijection between the odd-degree irreducible characters of |$G = {\sf S}_n$|⁠, |$GL_n(q)$| or |$GU_n(q)$| with |$q$| odd, and those of |${\mathbf N}_{G}(P)$|⁠, where |$P$| is a Sylow |$2$|-subgroup of |$G$|⁠. Since our bijections commute with the action of the absolute Galois group over the rationals, we conclude that the fields of values of character correspondents are the same. We use this to answer some questions of R. Gow.

On ∗-clean group rings over abelian groups

An associative ring with unity is called clean if each of its elements is the sum of an idempotent and a unit. A clean ring with involution ∗ is called ∗-clean if each of its elements is the sum of a unit and a projection (∗-invariant idempotent). In a recent paper, Huang, Li and Yuan provided a complete characterization that when a group ring [Formula: see text] is ∗-clean, where [Formula: see text] is a finite field and [Formula: see text] is a cyclic group of an odd prime power order [Formula: see text]. They also provided a necessary condition and a few sufficient conditions for [Formula: see text] to be ∗-clean, where [Formula: see text] is a cyclic group of order [Formula: see text]. In this paper, we extend the above result of Huang, Li and Yuan from [Formula: see text] to [Formula: see text] and provide a characterization of ∗-clean group rings [Formula: see text], where [Formula: see text] is a finite abelian group and [Formula: see text] is a field with characteristic not dividing the exponent of [Formula: see text].

The Gelfand-Graev Representation of GSp(4, 𝔽q)

ABSTRACTIn this article, we demonstrate a technique for calculating the irreducible representations of the endomorphism algebra of the Gelfand-Graev representations of the general symplectic group GSp(4, 𝔽q), where q is an odd prime power.

Cayley digraphs of 2-genetic groups of odd prime-power order

A group is called 2-genetic if each normal subgroup of the group can be generated by two elements. Let G be a non-abelian 2-genetic group of order pn for an odd prime p and a positive integer n. In this paper, we investigate connected Cayley digraphs Cay(G,S) for non-abelian 2-genetic groups G of odd order pn, and determine their full automorphism groups A=Aut(Cay(G,S)) in the case when Aut(G,S)={α∈Aut(G)|Sα=S} is a p′-group. It is shown that either Cay(G,S) is normal, that is, the right regular representation of G is normal in A, or p=3,5,7,11 and the largest normal p-subgroup Op(A) of A has order pn−1 with ASL(2,p)≤A/Φ(Op(A))≤AGL(2,p). Furthermore, a non-normal Cayley digraph with smallest order and smallest valency is constructed for each p=3,5,7,11, respectively. In particular, the underlying graphs of the above non-normal Cayley digraphs for p=3,7,11 are half-arc-transitive, and they are the first constructions of half-arc-transitive non-normal Cayley graphs of order a prime-power.

Incidences between points and generalized spheres over finite fields and related problems

Abstract Let 𝔽 q ${\mathbb{F}_{q}}$ be a finite field of q elements, where q is a large odd prime power and Q = a 1 ⁢ x 1 c 1 + ⋯ + a d ⁢ x d c d ∈ 𝔽 q ⁢ [ x 1 , … , x d ] , ${Q=a_{1}x_{1}^{c_{1}}+\cdots+a_{d}x_{d}^{c_{d}}\in\mathbb{F}_{q}[x_{1},\ldots,% x_{d}]},$ where 2 ≤ c i ≤ N ${2\leq c_{i}\leq N}$ , gcd ⁡ ( c i , q ) = 1 ${\gcd(c_{i},q)=1}$ , and a i ∈ 𝔽 q ${a_{i}\in\mathbb{F}_{q}}$ for all 1 ≤ i ≤ d ${1\leq i\leq d}$ . A Q -sphere is a set of the form { 𝒙 ∈ 𝔽 q d ∣ Q ⁢ ( 𝒙 - 𝒃 ) = r } , ${\bigl{\{}\boldsymbol{x}\in\mathbb{F}_{q}^{d}\mid Q(\boldsymbol{x}-\boldsymbol% {b})=r\bigr{\}}},$ where 𝒃 ∈ 𝔽 q d , r ∈ 𝔽 q ${\boldsymbol{b}\in\mathbb{F}_{q}^{d},r\in\mathbb{F}_{q}}$ . We prove bounds on the number of incidences between a point set 𝒫 ${{{\mathcal{P}}}}$ and a Q-sphere set 𝒮 ${{{\mathcal{S}}}}$ , denoted by I ⁢ ( 𝒫 , 𝒮 ) ${I({{\mathcal{P}}},{{\mathcal{S}}})}$ , as the following: | I ⁢ ( 𝒫 , 𝒮 ) - | 𝒫 | ⁢ | 𝒮 | q | ≤ q d / 2 ⁢ | 𝒫 | ⁢ | 𝒮 | . $\Biggl{|}I({{\mathcal{P}}},{{\mathcal{S}}})-\frac{|{{\mathcal{P}}}||{{\mathcal% {S}}}|}{q}\Biggr{|}\leq q^{d/2}\sqrt{|{{\mathcal{P}}}||{{\mathcal{S}}}|}.$ We also give a version of this estimate over finite cyclic rings ℤ / q ⁢ ℤ ${\mathbb{Z}/q\mathbb{Z}}$ , where q is an odd integer. As a consequence of the above bounds, we give an estimate for the pinned distance problem and a bound on the number of incidences between a random point set and a random Q-sphere set in 𝔽 q d ${\mathbb{F}_{q}^{d}}$ . We also study the finite field analogues of some combinatorial geometry problems, namely, the number of generalized isosceles triangles, and the existence of a large subset without repeated generalized distances.

Complete weight enumerators of some linear codes from quadratic forms

Linear codes with few weights have applications in secrete sharing, authentication codes, association schemes, and strongly regular graphs. In this paper, a construction of q-ary linear codes with few weights employing general quadratic forms over the finite field Fq${\mathbb {F}}_{q}$ is proposed, where q is an odd prime power. This generalizes some earlier constructions of p-ary linear codes from quadratic bent functions over the prime field Fp${\mathbb {F}}_{p}$, where p is an odd prime. The complete weight enumerators of the resultant q-ary linear codes are also determined.

Covariant mutually unbiased bases

The connection between maximal sets of mutually unbiased bases (MUBs) in a prime-power dimensional Hilbert space and finite phase-space geometries is well known. In this article, we classify MUBs according to their degree of covariance with respect to the natural symmetries of a finite phase-space, which are the group of its affine symplectic transformations. We prove that there exist maximal sets of MUBs that are covariant with respect to the full group only in odd prime-power dimensional spaces, and in this case, their equivalence class is actually unique. Despite this limitation, we show that in dimension [Formula: see text] covariance can still be achieved by restricting to proper subgroups of the symplectic group, that constitute the finite analogues of the oscillator group. For these subgroups, we explicitly construct the unitary operators yielding the covariance.

Permutation Symmetry Determines the Discrete Wigner Function.

The Wigner function provides a useful quasiprobability representation of quantum mechanics, with applications in various branches of physics. Many nice properties of the Wigner function are intimately connected with the high symmetry of the underlying operator basis composed of phase point operators: any pair of phase point operators can be transformed to any other pair by a unitary symmetry transformation. We prove that, in the discrete scenario, this permutation symmetry is equivalent to the symmetry group being a unitary 2 design. Such a highly symmetric representation can only appear in odd prime power dimensions besides dimensions 2 and 8. It suffices to single out a unique discrete Wigner function among all possible quasiprobability representations. In the course of our study, we show that this discrete Wigner function is uniquely determined by Clifford covariance, while no Wigner function is Clifford covariant in any even prime power dimension.

Sets with many pairs of orthogonal vectors over finite fields

Let n be a positive integer and B be a non-degenerate symmetric bilinear form over Fqn, where q is an odd prime power and Fq is the finite field with q elements. We determine the largest possible size of a subset S of Fqn such that |{B(x,y)|x,y∈S and x≠y}|=1. We also pose some conjectures concerning nearly orthogonal subsets of Fqn where a nearly orthogonal subset T of Fqn is a set of vectors in which among any three distinct vectors there are two vectors x, y so that B(x,y)=0.

Galois unitaries, mutually unbiased bases, and mub-balanced states

A Galois unitary is a generalization of the notion of anti-unitary operators. They act only on those vectors in Hilbert space whose entries belong to some chosen number field. For Mutually Unbiased Bases the relevant number field is a cyclotomic field. By including Galois unitaries we are able to remove a mismatch between the finite projective group acting on the bases on the one hand, and the set of those permutations of the bases that can be implemented as transformations in Hilbert space on the other hand. In particular we show that there exist transformations that cycle through all the bases in all dimensions d = p^n where p is an odd prime and the exponent n is odd. (For even primes unitary MUB-cyclers exist.) These transformations have eigenvectors, which are MUB-balanced states (i.e. rotationally symmetric states in the original terminology of Wootters and Sussman) if and only if d = 3 modulo 4. We conjecture that this construction yields all such states in odd prime power dimension.

A class of constacyclic codes over a finite field-II

Let \(\mathbb{F}_q\) be a finite field with q = pm elements, where p is any prime and m ≥ 1. In this paper, we explicitly determine all the μ-constacyclic codes of length ln over \(\mathbb{F}_q\), where l is an odd prime coprime to p and the order of μ is a power of l. All the repeated-root λ- constacyclic codes of length lnps over \(\mathbb{F}_q\) are also determined for any nonzero λ in \(\mathbb{F}_q\). As examples all the λ-constacyclic codes of length 3nps over \(\mathbb{F}_q\) for p = 5, 7, 11, 19 for n ≥ 1, s ≥ 1 are derived. We also obtain all the self-orthogonal negacyclic codes of length ln over \(\mathbb{F}_q\) when q is odd prime power and give some illustrative examples.