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Yue Zhou

Publications and source records attributed to Yue Zhou.

At least 199 records · Page 11Linked to original sources
arXiv2015

Unitals in shift planes of odd order

A finite shift plane can be equivalently defined via abelian relative difference sets as well as planar functions. In this paper, we present a generic way to construct unitals in finite shift planes of odd orders $q^2$. We investigate various geometric and combinatorial properties of them, such as the self-duality, the existences of O'Nan configurations, the Wilbrink's conditions, the designs formed by circles and so on. We also show that our unitals are inequivalent to the unitals derived from unitary polarities in the same shift planes. As designs, our unitals are also not isomorphic to the classical unitals (the Hermitian curves).

math.CO↗
arXiv2015

Cayley graphs of diameter two from difference sets

Let $C(d,k)$ and $AC(d,k)$ be the largest order of a Cayley graph and a Cayley graph based on an abelian group, respectively, of degree $d$ and diameter $k$. When $k=2$, it is well-known that $C(d,2)\le d^2+1$ with equality if and only if the graph is a Moore graph. In the abelian case, we have $AC(d,2)\le \frac{d^2}{2}+d+1$. The best currently lower bound on $AC(d,2)$ is $\frac{3}{8}d^2-1.45 d^{1.525}$ for all sufficiently large $d$. In this paper, we consider the construction of large graphs of diameter $2$ using generalized difference sets. We show that $AC(d,2)\ge \frac{25}{64}d^2-2.1 d^{1.525}$ for sufficiently large $d$ and $AC(d,2) \ge \frac{4}{9}d^2$ if $d=3q$, $q=2^m$ and $m$ is odd.

math.CO↗
arXiv2014

Terahertz Generation in Lithium Niobate Driven by Ti:Sapphire Laser Pulses and its Limitations

We experimentally investigate the limits to 800 nm-to-terahertz (THz) energy conversion in lithium niobate at room temperature driven by amplified Ti:Sapphire laser pulses with tilted-pulse-front. The influence of the pump central wavelength, pulse duration, and fluence on THz generation is studied. We achieved a high peak efficiency of 0.12% using transform limited 150 fs pulses and observed saturation of the optical to THz conversion efficiency at a fluence of 15 mJ/cm2. We experimentally identify two main limitations for the scaling of optical-to-THz conversion efficiencies: (i) the large spectral broadening of the optical pump spectrum in combination with large angular dispersion of the tilted-pulse-front and (ii) free-carrier absorption of THz radiation due to multi-photon absorption of the 800 nm radiation.

physics.optics↗
arXiv2014

Switchings of semifield multiplications

Let $B(X,Y)$ be a polynomial over $\mathbb{F}_{q^n}$ which defines an $\mathbb{F}_q$-bilinear form on the vector space $\mathbb{F}_{q^n}$, and let $ξ$ be a nonzero element in $\mathbb{F}_{q^n}$. In this paper, we consider for which $B(X,Y)$, the binary operation $xy+B(x,y)ξ$ defines a (pre)semifield multiplication on $\mathbb{F}_{q^n}$. We prove that this question is equivalent to finding $q$-linearized polynomials $L(X)\in\mathbb{F}_{q^n}[X]$ such that $Tr_{q^n/q}(L(x)/x)\neq 0$ for all $x\in\mathbb{F}_{q^n}^*$. For $n\le 4$, we present several families of $L(X)$ and we investigate the derived (pre)semifields. When $q$ equals a prime $p$, we show that if $n>\frac{1}{2}(p-1)(p^2-p+4)$, $L(X)$ must be $a_0 X$ for some $a_0\in\mathbb{F}_{p^n}$ satisfying $Tr_{q^n/q}(a_0)\neq 0$. Finally, we include a natural connection with certain cyclic codes over finite fields, and we apply the Hasse-Weil-Serre bound for algebraic curves to prove several necessary conditions for such kind of $L(X)$.

math.CO↗
arXiv2014

Exceptional planar polynomials

Planar functions are special functions from a finite field to itself that give rise to finite projective planes and other combinatorial objects. We consider polynomials over a finite field $K$ that induce planar functions on infinitely many extensions of $K$; we call such polynomials exceptional planar. Exceptional planar monomials have been recently classified. In this paper we establish a partial classification of exceptional planar polynomials. This includes results for the classical planar functions on finite fields of odd characteristic and for the recently proposed planar functions on finite fields of characteristic two.

math.NT↗
arXiv2014

Semifields, relative difference sets, and bent functions

Recently, the interest in semifields has increased due to the discovery of several new families and progress in the classification problem. Commutative semifields play an important role since they are equivalent to certain planar functions (in the case of odd characteristic) and to modified planar functions in even characteristic. Similarly, commutative semifields are equivalent to relative difference sets. The goal of this survey is to describe the connection between these concepts. Moreover, we shall discuss power mappings that are planar and consider component functions of planar mappings, which may be also viewed as projections of relative difference sets. It turns out that the component functions in the even characteristic case are related to negabent functions as well as to $\mathbb{Z}_4$-valued bent functions.

math.CO↗
arXiv2014

Planar functions over fields of characteristic two

Classical planar functions are functions from a finite field to itself and give rise to finite projective planes. They exist however only for fields of odd characteristic. We study their natural counterparts in characteristic two, which we also call planar functions. They again give rise to finite projective planes, as recently shown by the second author. We give a characterisation of planar functions in characteristic two in terms of codes over $\mathbb{Z}_4$. We then specialise to planar monomial functions $f(x)=cx^t$ and present constructions and partial results towards their classification. In particular, we show that $t=1$ is the only odd exponent for which $f(x)=cx^t$ is planar (for some nonzero $c$) over infinitely many fields. The proof techniques involve methods from algebraic geometry.

math.CO↗
arXiv2013

Sets of Zero-Difference Balanced Functions and Their Applications

Zero-difference balanced (ZDB) functions can be employed in many applications, e.g., optimal constant composition codes, optimal and perfect difference systems of sets, optimal frequency hopping sequences, etc. In this paper, two results are summarized to characterize ZDB functions, among which a lower bound is used to achieve optimality in applications and determine the size of preimage sets of ZDB functions. As the main contribution, a generic construction of ZDB functions is presented, and many new classes of ZDB functions can be generated. This construction is then extended to construct a set of ZDB functions, in which any two ZDB functions are related uniformly. Furthermore, some applications of such sets of ZDB functions are also introduced.

cs.IT↗
arXiv2013

(2^n,2^n,2^n,1)-relative difference sets and their representations

We show that every $(2^n,2^n,2^n,1)$-relative difference set $D$ in $\Z_4^n$ relative to $\Z_2^n$ can be represented by a polynomial $f(x)\in \F_{2^n}[x]$, where $f(x+a)+f(x)+xa$ is a permutation for each nonzero $a$. We call such an $f$ a planar function on $\F_{2^n}$. The projective plane $Π$ obtained from $D$ in the way of Ganley and Spence \cite{ganley_relative_1975} is coordinatized, and we obtain necessary and sufficient conditions of $Π$ to be a presemifield plane. We also prove that a function $f$ on $\F_{2^n}$ with exactly two elements in its image set and $f(0)=0$ is planar, if and only if, $f(x+y)=f(x)+f(y)$ for any $x,y\in\F_{2^n}$.

math.CO↗
arXiv2012

A Note on the Isotopism of Commutative Semifields

We present an example of two isotopic but not strongly isotopic commutative semifields. This example shows that a recent result of Coulter and Henderson on semifield of order p^n, n odd, can not be generalized to the case n even.

math.CO↗
arXiv2012

New extensions to the sumsets with polynomial restrictions

By taking the leading and the second leading coefficients of the Morris identity, we get new polynomial coefficients. These coefficients lead to new results in the sumsets with polynomial restrictions by the polynomial method of N. Alon.

math.CO↗
arXiv2011

A new family of semifields with 2 parameters

A new family of commutative semifields with two parameters is presented. Its left and middle nucleus are both determined. Furthermore, we prove that for any different pairs of parameters, these semifields are not isotopic. It is also shown that, for some special parameters, one semifield in this family can lead to two inequivalent planar functions. Finally, using similar construction, new APN functions are given.

math.CO↗
arXiv2011

Optimization of Fast-Decodable Full-Rate STBC with Non-Vanishing Determinants

Full-rate STBC (space-time block codes) with non-vanishing determinants achieve the optimal diversity-multiplexing tradeoff but incur high decoding complexity. To permit fast decoding, Sezginer, Sari and Biglieri proposed an STBC structure with special QR decomposition characteristics. In this paper, we adopt a simplified form of this fast-decodable code structure and present a new way to optimize the code analytically. We show that the signal constellation topology (such as QAM, APSK, or PSK) has a critical impact on the existence of non-vanishing determinants of the full-rate STBC. In particular, we show for the first time that, in order for APSK-STBC to achieve non-vanishing determinant, an APSK constellation topology with constellation points lying on square grid and ring radius $\sqrt{m^2+n^2} (m,n\emph{\emph{integers}})$ needs to be used. For signal constellations with vanishing determinants, we present a methodology to analytically optimize the full-rate STBC at specific constellation dimension.

cs.IT↗
arXiv2010

On Kadell's two Conjectures for the $q$-Dyson Product

By extending Lv-Xin-Zhou's first layer formulas of the $q$-Dyson product, we prove Kadell's conjecture for the Dyson product and show the error of his $q$-analogous conjecture. With the extended formulas we establish a $q$-analog of Kadell's conjecture for the Dyson product.

math.CO↗
arXiv2009

Optimal teleportation via thermal entangled states of a two-qubit Heisenberg Chain

We study the optimal teleportation based on Bell measurements via the thermal states of a two-qubit Heisenberg XXX chain in the presence of Dzyaloshinsky-Moriya (DM) anisotropic antisymmetric interaction and obtain the optimal unitary transformation. The explicit expressions of the output state and the teleportation fidelity are presented and compared with those of the standard protocol. It is shown that in this protocol the teleportation fidelity is always larger and unit fidelity is achieved at zero temperature. The DM interaction can enhance the teleportation fidelity at finite temperatures, as opposed to the effect of the interaction in the standard protocol. Cases with other types of anisotropies are also discussed.

quant-ph↗
arXiv2008

Geometric phase of a bipartite system with Dzyaloshinski-Moriya interaction

The Berry phase of a bipartite system described by a Heisenberg XXZ model driven by a one-site magnetic field is investigated. The effect of the Dzyaloshinski-Moriya (DM) anisotropic interaction on the Berry phase is discussed. It is found that the DM interaction affects the Berry phase monotonously, and can also cause sudden change of the Berry phase for some weak magnetic field cases.

quant-ph↗