arXiv subjects

Yue Zhou

Publications and source records attributed to Yue Zhou.

At least 181 records · Page 10Linked to original sources

arXiv2020

A recursion for a symmetric function generalization of the $q$-Dyson constant term identity

In 2000, Kadell gave an orthogonality conjecture for a symmetric function generalization of the $q$-Dyson constant term identity or the Zeilberger--Bressoud $q$-Dyson theorem. The non-zero part of Kadell's orthogonality conjecture is a constant term identity indexed by a weak composition $v=(v_1,\dots,v_n)$ in the case when only one $v_i\neq 0$. This conjecture was first proved by Károlyi, Lascoux and Warnaar in 2015. They further formulated a closed-form expression for the above mentioned constant term in the case when all the parts of $v$ are distinct. Recently we obtain a recursion for this constant term provided that the largest part of $v$ occurs with multiplicity one in $v$. In this paper, we generalize our previous result to all compositions $v$.

math.CO↗

arXiv2020

A lower bound on the number of inequivalent APN functions

In this paper, we establish a lower bound on the total number of inequivalent APN functions on the finite field with $2^{2m}$ elements, where $m$ is even. We obtain this result by proving that the APN functions introduced by Pott and the second author, that depend on three parameters $k$, $s$ and $α$, are pairwise inequivalent for distinct choices of the parameters $k$ and $s$. Moreover, we determine the automorphism group of these APN functions.

math.CO↗

arXiv2020

Rational Kernel on Pricing Models of Inflation Derivatives

The aim of this thesis is to analyze and renovate few main-stream models on inflation derivatives. In the first chapter of the thesis, concepts of financial instruments and fundamental terms are introduced, such as coupon bond, inflation-indexed bond, swap. In the second chapter of the thesis, classic models along the history of developing quantified interest rate models are introduced and analyzed. Moreover, the classification of interest rate models is introduced to help audiences understand the intrinsic ideology behind each type of models. In the third chapter of the thesis, the related mathematical knowledge is introduced. This part has the contribution on understanding the terms and relation among terms in each model introduced previously. In the fourth part of the thesis, the renovation of HJM frame work is introduced and analysis has been initiated.

q-fin.MF↗

arXiv2019

On the $q$-Dyson orthogonality problem

By combining the Gessel--Xin method with plethystic substitutions, we obtain a recursion for a symmetric function generalization of the $q$-Dyson constant term identity also known as the Zeilberger--Bressoud $q$-Dyson theorem. This yields a constant term identity which generalizes the non-zero part of Kadell's orthogonality ex-conjecture and a result of Károlyi, Lascoux and Warnaar.

math.CO↗

arXiv2019

New lower bounds for partial $k$-parallelisms

Due to the applications in network coding, subspace codes and designs have received many attentions. Suppose that $k\mid n$ and $V(n,q)$ is an $n$-dimensional space over the finite field $\mathbb{F}_{q}$. A $k$-spread is a $\frac{q^n-1}{q^k-1}$-set of $k$-dimensional subspaces of $V(n,q)$ such that each nonzero vector is covered exactly once. A partial $k$-parallelism in $V(n,q)$ is a set of pairwise disjoint $k$-spreads. As the number of $k$-dimensional subspaces in $V(n,q)$ is ${n \brack k}_{q}$, there are at most ${n-1 \brack k-1}_{q}$ spreads in a partial $k$-parallelism. By studying the independence numbers of Cayley graphs associated to a special type of partial $k$-parallelisms in $V(n,q)$, we obtain new lower bounds for partial $k$-parallelisms. In particular, we show that there exist at least $\frac{q^{k}-1}{q^{n}-1}{n-1 \brack k-1}_q$ pairwise disjoint $k$-spreads in $V(n,q)$.

math.CO↗

arXiv2019

No lattice tiling of $\mathbb{Z}^n$ by Lee Sphere of radius 2

We prove the nonexistence of lattice tilings of $\mathbb{Z}^n$ by Lee spheres of radius $2$ for all dimensions $n\geq 3$. This implies that the Golomb-Welch conjecture is true when the common radius of the Lee spheres equals $2$ and $2n^2+2n+1$ is a prime. As a direct consequence, we also answer an open question in the degree-diameter problem of graph theory: the order of any abelian Cayley graph of diameter $2$ and degree larger than $5$ cannot meet the abelian Cayley Moore bound.

math.CO↗

arXiv2019

Nuclei and automorphism groups of generalized twisted Gabidulin codes

Generalized twisted Gabidulin codes are one of the few known families of maximum rank matrix codes over finite fields. As a subset of m by n matrices, when m=n, the automorphism group of any generalized twisted Gabidulin code has been completely determined recently. In this paper, we consider the same problem for m<n. Under certain conditions on their parameters, we determine their middle nuclei and right nuclei, which are important invariants with respect to the equivalence for rank metric codes. Furthermore, we also use them to derive necessary conditions on the automorphisms of generalized twisted Gabidulin codes.

math.CO↗

arXiv2019

On the nonexistence of linear perfect Lee codes

In 1968, Golomb and Welch conjectured that there does not exist perfect Lee code in $\mathbb{Z}^{n}$ with radius $r\ge2$ and dimension $n\ge3$. Besides its own interest in coding theory and discrete geometry, this conjecture is also strongly related to the degree-diameter problems of abelian Cayley graphs. Although there are many papers on this topic, the Golomb-Welch conjecture is far from being solved. In this paper, we prove the nonexistence of linear perfect Lee codes by introducing some new algebraic methods. Using these new methods, we show the nonexistence of linear perfect Lee codes of radii $r=2,3$ in $\mathbb{Z}^n$ for infinitely many values of the dimension $n$. In particular, there does not exist linear perfect Lee codes of radius $2$ in $\mathbb{Z}^n$ for all $3\le n\le 100$ except 8 cases.

math.CO↗

arXiv2019

A new family of MRD codes in $\mathbb F_q^{2n\times2n}$ with right and middle nuclei $\mathbb F_{q^n}$

In this paper, we present a new family of maximum rank distance (MRD for short) codes in $\mathbb F_{q}^{2n\times 2n}$ of minimum distance $2\leq d\leq 2n$. In particular, when $d=2n$, we can show that the corresponding semifield is exactly a Hughes-Kleinfeld semifield. The middle and right nuclei of these MRD codes are both equal to $\mathbb F_{q^n}$. We also prove that the MRD codes of minimum distance $2<d<2n$ in this family are inequivalent to all known ones. The equivalence between any two members of this new family is also determined.

math.CO↗

arXiv2018

Efficient spectrum prediction and inverse design for plasmonic waveguide systems based on artificial neural networks

In this article, we propose a novel approach to achieve spectrum prediction, parameter fitting, inverse design and performance optimization for the plasmonic waveguide coupled with cavities structure (PWCCS) based on artificial neural networks (ANNs). The Fano resonance and plasmon induced transparency effect originated from the PWCCS have been selected as illustrations to verify the effectiveness of ANNs. We use the genetic algorithm to design the network architecture and select the hyper-parameters for ANNs. Once ANNs are trained by using a small sampling of the data generated by Monte Carlo method, the transmission spectrums predicted by the ANNs are quite approximate to the simulated results. The physical mechanisms behind the phenomena are discussed theoretically, and the uncertain parameters in the theoretical models are fitted by utilizing the trained ANNs. More importantly, our results demonstrate that this model-driven method not only realizes the inverse design of the PWCCS with high precision but also optimizes some critical performance metrics for transmission spectrum. Compared with previous works, we construct a novel model-driven analysis method for the PWCCS which are expected to have significant applications in the device design, performance optimization, variability analysis, defect detection, theoretical modeling, optical interconnects and so on.

physics.optics↗

arXiv2018

Generalized Twisted Gabidulin Codes

Let $\mathcal{C}$ be a set of $m$ by $n$ matrices over $\mathbb{F}_q$ such that the rank of $A-B$ is at least $d$ for all distinct $A,B\in \mathcal{C}$. Suppose that $m\leqslant n$. If $\#\mathcal{C}= q^{n(m-d+1)}$, then $\mathcal{C}$ is a maximum rank distance (MRD for short) code. Until 2016, there were only two known constructions of MRD codes for arbitrary $1<d<m-1$. One was found by Delsarte (1978) and Gabidulin (1985) independently, and it was later generalized by Kshevetskiy and Gabidulin (2005). We often call them (generalized) Gabidulin codes. Another family was recently obtained by Sheekey (2016), and its elements are called twisted Gabidulin codes. In the same paper, Sheekey also proposed a generalization of the twisted Gabidulin codes. However the equivalence problem for it is not considered, whence it is not clear whether there exist new MRD codes in this generalization. We call the members of this putative larger family generalized twisted Gabidulin codes. In this paper, we first compute the Delsarte duals and adjoint codes of them, then we completely determine the equivalence between different generalized twisted Gabidulin codes. In particular, it can be proven that, up to equivalence, generalized Gabidulin codes and twisted Gabidulin codes are both proper subsets of this family.

math.CO↗

arXiv2018

Polarization Weight Family Methods for Polar Code Construction

Polar codes are the first proven capacity-achieving codes. Recently, they are adopted as the channel coding scheme for 5G due to their superior performance.A polar code for encoding length-K information bits in length-N codeword could be specified by the polar code construction method. Most construction methods define a polar code related to channel parameter set, e.g. designed signal-to-noise ratio. Polarization weight (PW) is a channel-independent approximation method, which estimates the subchannel reliability as a function of its index. In this paper, we generalize the PW method by including higher-order bases or extended bases. The proposed methods have robust performance while preserving the computational and mathematical simplicity as PW.

cs.IT↗

arXiv2018

On the number of inequivalent Gabidulin codes

Maximum rank-distance (MRD) codes are extremal codes in the space of $m\times n$ matrices over a finite field, equipped with the rank metric. Up to generalizations, the classical examples of such codes were constructed in the 1970s and are today known as Gabidulin codes. Motivated by several recent approaches to construct MRD codes that are inequivalent to Gabidulin codes, we study the equivalence issue for Gabidulin codes themselves. This shows in particular that the family of Gabidulin codes already contains a huge subset of MRD codes that are pairwise inequivalent, provided that $2\le m\le n-2$.

math.CO↗

arXiv2017

Exceptional Scattered Polynomials

Let $f$ be an $\mathbb{F}_q$-linear function over $\mathbb{F}_{q^n}$. If the $\mathbb{F}_q$-subspace $U= \{ (x^{q^t}, f(x)) : x\in \mathbb{F}_{q^n} \}$ defines a maximum scattered linear set, then we call $f$ a scattered polynomial of index $t$. As these polynomials appear to be very rare, it is natural to look for some classification of them. We say a function $f$ is an exceptional scattered polynomial of index $t$ if the subspace $U$ associated with $f$ defines a maximum scattered linear set in $\mathrm{PG}(1, q^{mn})$ for infinitely many $m$. Our main results are the complete classifications of exceptional scattered monic polynomials of index $0$ (for $q>5$) and of index $1$. The strategy applied here is to convert the original question into a special type of algebraic curves and then to use the intersection theory and the Hasse-Weil theorem to derive contradictions.

math.CO↗

arXiv2017

Frequency-oriented sub-sampling by photonic Fourier transform and I/Q demodulation

Sub-sampling can acquire directly a passband within a broad radio frequency (RF) range, avoiding down-conversion and low-phase-noise tunable local oscillation (LO). However, sub-sampling suffers from band folding and self-image interference. In this paper we propose a frequency-oriented sub-sampling to solve the two problems. With ultrashort optical pulse and a pair of chromatic dispersions, the broadband RF signal is firstly short-time Fourier-transformed to a spectrum-spread pulse. Then a time slot, corresponding to the target spectrum slice, is coherently optical-sampled with in-phase/quadrature (I/Q) demodulation. We demonstrate the novel bandpass sampling by a numerical example, which shows the desired uneven intensity response, i.e. pre-filtering. We show in theory that appropriate time-stretch capacity from dispersion can result in pre-filtering bandwidth less than sampling rate. Image rejection due to I/Q sampling is also analyzed. A proof-of-concept experiment, which is based on a time-lens sampling source and chirped fiber Bragg gratings (CFBGs), shows the center-frequency-tunable pre-filtered sub-sampling with bandwidth of 6 GHz around, as well as imaging rejection larger than 26 dB. Our technique may benefit future broadband RF receivers for frequency-agile Radar or channelization.

physics.ins-det↗

arXiv2017

On kernels and nuclei of rank metric codes

For each rank metric code $\mathcal{C}\subseteq \mathbb{K}^{m\times n}$, we associate a translation structure, the kernel of which is shown to be invariant with respect to the equivalence on rank metric codes. When $\mathcal{C}$ is $\mathbb{K}$-linear, we also propose and investigate other two invariants called its middle nucleus and right nucleus. When $\mathbb{K}$ is a finite field $\mathbb{F}_q$ and $\mathcal{C}$ is a maximum rank distance code with minimum distance $d<\min\{m,n\}$ or $\gcd(m,n)=1$, the kernel of the associated translation structure is proved to be $\mathbb{F}_q$. Furthermore, we also show that the middle nucleus of a linear maximum rank distance code over $\mathbb{F}_q$ must be a finite field; its right nucleus also has to be a finite field under the condition $\max\{d,m-d+2\} \geqslant \left\lfloor \frac{n}{2} \right\rfloor +1$. Let $\mathcal{D}$ be the DHO-set associated with a bilinear dimensional dual hyperoval over $\mathbb{F}_2$. The set $\mathcal{D}$ gives rise to a linear rank metric code, and we show that its kernel and right nucleus are is isomorphic to $\mathbb{F}_2$. Also, its middle nucleus must be a finite field containing $\mathbb{F}_q$. Moreover, we also consider the kernel and the nuclei of $\mathcal{D}^k$ where $k$ is a Knuth operation.

math.CO↗

arXiv2016

Hyperovals in Knuth's binary semifield planes

In each of the three projective planes coordinatised by the Knuth's binary semifield $\mathbb{K}_n$ of order $2^n$ and two of its Knuth derivatives, we exhibit a new family of infinitely many translation hyperovals. In particular, when $n=5$, we also present complete lists of all translation hyperovals in them. The properties of some designs associated with these hyperovals are also studied.

math.CO↗

arXiv2015

On the 2-ranks of a class of unitals

Let $U_θ$ be a unital defined in a shift plane of odd order $q^2$, which are constructed recently by the authors. In particular, when the shift plane is desarguesian, $U_θ$ is a special Buekenhout-Metz unital formed by a union of ovals. We investigate the dimensions of the binary codes derived from $U_θ$. By using Kloosterman sums, we obtain a new lower bound on the aforementioned dimensions which improves the result obtained by Leung and Xiang in 2009. In particular, for $q=3^m$, this new lower bound equals $\frac{2}{3}(q^3+q^2-2q)-1$ for even $m$ and $\frac{2}{3}(q^3+q^2+q)-1$ for odd $m$.

math.CO↗