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Qing Xiang

Publications and source records attributed to Qing Xiang.

At least 55 records · Page 3Linked to original sources
arXiv2011

Partial difference sets from quadratic forms and $p$-ary weakly regular bent functions

We generalize the construction of affine polar graphs in two different ways to obtain new partial difference sets and amorphic association schemes. The first generalization uses a combination of quadratic forms and uniform cyclotomy. In the second generalization we replace the quadratic form in the affine polar graph construction by higher degree homogeneous functions that are $p$-ary weakly regular bent. The negative Latin square type partial difference sets arising from the first generalization are new.

math.CO↗
arXiv2011

Exterior algebras and two conjectures on finite abelian groups

Let G be a finite abelian group with |G|>1. Let a_1,...,a_k be k distinct elements of G and let b_1,...,b_k be (not necessarily distinct) elements of G, where k is a positive integer smaller than the least prime divisor of |G|. We show that there is a permutation $π$ on {1,...,k} such that a_1b_{π(1)},...,a_kb_{π(k)} are distinct, provided that any other prime divisor of |G| (if there is any) is greater than k!. This in particular confirms the Dasgupta-Karolyi-Serra-Szegedy conjecture for abelian p-groups. We also pose a new conjecture involving determinants and characters, and show that its validity implies Snevily's conjecture for abelian groups of odd order. Our methods involve exterior algebras and characters.

math.GR↗
arXiv2010

Permutation Polynomials of Degree 6 or 7 over Finite Fields of Characteristic 2

In \cite{D1}, Dickson listed all permutation polynomials up to degree 5 over an arbitrary finite field, and all permutation polynomials of degree 6 over finite fields of odd characteristic. The classification of degree 6 permutation polynomials over finite fields of characteristic 2 was left incomplete. In this paper we complete the classification of permutation polynomials of degree 6 over finite fields of characteristic 2. In addition, all permutation polynomials of degree 7 over finite fields of characteristic 2 are classified.

math.CO↗
arXiv2009

Dimensions of Some Binary Codes Arising From A Conic in $PG(2,q)$

Let $\mathcal{O}$ be a conic in the classical projective plane $PG(2,q)$, where $q$ is an odd prime power. With respect to $\mathcal{O}$, the lines of $PG(2,q)$ are classified as passant, tangent, and secant lines, and the points of $PG(2,q)$ are classified as internal, absolute and external points. The incidence matrices between the secant/passant lines and the external/internal points were used in \cite{keith1} to produce several classes of structured low-density parity-check binary codes. In particular, the authors of \cite{keith1} gave conjectured dimension formula for the binary code $\mathcal{L}$ which arises as the $\Ff_2$-null space of the incidence matrix between the secant lines and the external points to $\mathcal{O}$. In this paper, we prove the conjecture on the dimension of $\mathcal{L}$ by using a combination of techniques from finite geometry and modular representation theory.

math.CO↗
arXiv2008

Proofs of two conjectures on ternary weakly regular bent functions

We study ternary monomial functions of the form $f(x)=\Tr_n(ax^d)$, where $x\in \Ff_{3^n}$ and $\Tr_n: \Ff_{3^n}\to \Ff_3$ is the absolute trace function. Using a lemma of Hou \cite{hou}, Stickelberger's theorem on Gauss sums, and certain ternary weight inequalities, we show that certain ternary monomial functions arising from \cite{hk1} are weakly regular bent, settling a conjecture of Helleseth and Kholosha \cite{hk1}. We also prove that the Coulter-Matthews bent functions are weakly regular.

math.CO↗
arXiv2008

Incidence Modules for Symplectic Spaces in Characteristic Two

We study the permutation action of a finite symplectic group of characteristic 2 on the set of subspaces of its standard module which are either totally isotropic or else complementary to totally isotropic subspaces with respect to the alternating form. A general formula is obtained for the 2-rank of the incidence matrix for the inclusion of one-dimensional subspaces in the distinguished subspaces of a fixed dimension.

math.CO↗
arXiv2008

Semi-regular Relative Difference Sets with Large Forbidden Subgroups

Motivated by a connection between semi-regular relative difference sets and mutually unbiased bases, we study relative difference sets with parameters $(m,n,m,m/n)$ in groups of non-prime-power orders. Let $p$ be an odd prime. We prove that there does not exist a $(2p,p,2p,2)$ relative difference set in any group of order $2p^2$, and an abelian $(4p,p,4p,4)$ relative difference set can only exist in the group $\Bbb{Z}_2^2\times \Bbb{Z}_3^2$. On the other hand, we construct a family of non-abelian relative difference sets with parameters $(4q,q,4q,4)$, where $q$ is an odd prime power greater than 9 and $q\equiv 1$ (mod 4). When $q=p$ is a prime, $p>9$, and $p\equiv$ 1 (mod 4), the $(4p,p,4p,4)$ non-abelian relative difference sets constructed here are genuinely non-abelian in the sense that there does not exist an abelian relative difference set with the same parameters.

math.CO↗
arXiv2007

On the dimensions of certain LDPC codes based on q-regular bipartite graphs

An explicit construction of a family of binary LDPC codes called LU(3,q), where q is a power of a prime, was recently given. A conjecture was made for the dimensions of these codes when q is odd. The conjecture is proved in this note. The proof involves the geometry of a 4-dimensional symplectic vector space and the action of the symplectic group and its subgroups.

cs.IT↗
arXiv2006

Skew Hadamard difference sets from the Ree-Tits slice symplectic spreads in PG(3,3^{2h+1})

Using a class of permutation polynomials of $F_{3^{2h+1}}$ obtained from the Ree-Tits symplectic spreads in $PG(3,3^{2h+1})$, we construct a family of skew Hadamard difference sets in the additive group of $F_{3^{2h+1}}$. With the help of a computer, we show that these skew Hadamard difference sets are new when $h=2$ and $h=3$. We conjecture that they are always new when $h>3$. Furthermore, we present a variation of the classical construction of the twin prime power difference sets, and show that inequivalent skew Hadamard difference sets lead to inequivalent difference sets with twin prime power parameters.

math.CO↗
arXiv2006

The permutation action of finite symplectic groups of odd characteristic on their standard modules

Motivated by the incidence problems between points and flats of a symplectic polar space, we study a large class of submodules of the space of functions on the standard module of a finite symplectic group of odd characteristic. Our structure results on this class of submodules allow us to determine the $p$-ranks of the incidence matrices between points and flats of the symplectic polar space. In particular, we give an explicit formula for the $p$-rank of the generalized quadrangle ${\rm W}(3,q)$, where $q$ is an odd prime power. Combined with the earlier results of Sastry and Sin on the 2-rank of ${\rm W}(3,2^t)$, it completes the determination of the $p$-ranks of ${\rm W}(3,q)$.

math.CO↗
arXiv2005

Pseudocyclic association schemes arising from the actions of PGL(2,2^m) and PΓL(2,2^m)

The action of $PGL(2,2^m)$ on the set of exterior lines to a nonsingular conic in $PG(2,2^m)$ affords an association scheme, which was shown to be pseudocyclic in Hollmann's thesis in 1982. It was further conjectured in Hollmann's thesis that the orbital scheme of $PΓL(2,2^m)$ on the set of exterior lines to a nonsingular conic in $PG(2,2^m)$ is also pseudocyclic if $m$ is an odd prime. We confirm this conjecture in this paper. As a by-product, we obtain a class of Latin square type strongly regular graphs on nonprime-power number of points.

math.CO↗
arXiv2005

Association schemes from the action of $PGL(2,q)$ fixing a nonsingular conic in PG(2,q)

The group $PGL(2,q)$ has an embedding into $PGL(3,q)$ such that it acts as the group fixing a nonsingular conic in $PG(2,q)$. This action affords a coherent configuration $R(q)$ on the set $L(q)$ of non-tangent lines of the conic. We show that the relations can be described by using the cross-ratio. Our results imply that the restrictions $R_{+}(q)$ and $R_{-}(q)$ to the sets $L_{+}(q)$ of secant lines and to the set $L_{-}(q)$ of exterior lines, respectively, are both association schemes; moreover, we show that the elliptic scheme $R_{-}(q)$ is pseudocyclic. We further show that the coherent configuration $R(q^2)$ with $q$ even allow certain fusions. These provide a 4-class fusion of the hyperbolic scheme $R_{+}(q^2)$, and 3-class fusions and 2-class fusions (strongly regular graphs) of both schemes $R_{+}(q^2)$ and $R_{-}(q^2). The fusion results for the hyperbolic case are known, but our approach here as well as our results in the elliptic case are new.

math.CO↗
arXiv2004

Amorphic association schemes with negative Latin square type graphs

Applying results from partial difference sets, quadratic forms, and recent results of Brouwer and Van Dam, we construct the first known amorphic association scheme with negative Latin square type graphs and whose underlying set is a non-elementary abelian 2-group. We give a simple proof of a result of Hamilton that generalizes Brouwer's result. We use multiple distinct quadratic forms to construct amorphic association schemes with a large number of classes.

math.CO↗
arXiv2004

Negative Latin square type partial difference sets in nonelementary abelian 2-groups

Combining results on quadrics in projective geometries with an algebraic interplay between finite fields and Galois rings, we construct the first known family of partial difference sets with negative Latin square type parameters in nonelementary abelian groups, the groups $\Z_4^{2k}\times \Z_2^{4 \ell-4k}$ for all $k$ when $\ell$ is odd and for all $k < \ell$ when $\ell$ is even. Similarly, we construct partial difference sets with Latin square type parameters in the same groups for all $k$ when $\ell$ is even and for all $k<\ell$ when $\ell$ is odd. These constructions provide the first example that the non-homomorphic bijection approach outlined by Hagita and Schmidt \cite{hagitaschmidt} can produce difference sets in groups that previously had no known constructions. Computer computations indicate that the strongly regular graphs associated to the PDSs are not isomorphic to the known graphs, and we conjecture that the family of strongly regular graphs will be new.

math.CO↗
arXiv2004

On Mathon's construction of maximal arcs in Desarguesian planes. II

In a recent paper [M], Mathon gives a new construction of maximal arcs which generalizes the construction of Denniston. In relation to this construction, Mathon asks the question of determining the largest degree of a non-Denniston maximal arc arising from his new construction. In this paper, we give a nearly complete answer to this problem. Specifically, we prove that when $m\geq 5$ and $m\neq 9$, the largest $d$ of a non-Denniston maximal arc of degree $2^d$ in PG(2,2^m) generated by a {p,1}-map is $(\floor {m/2} +1)$. This confirms our conjecture in [FLX]. For {p,q}-maps, we prove that if $m\geq 7$ and $m\neq 9$, then the largest $d$ of a non-Denniston maximal arc of degree $2^d$ in PG(2,2^m) generated by a {p,q}-map is either $\floor {m/2} +1$ or $\floor{m/2} +2$.

math.CO↗