# Shuxing LiUniversity of Delaware | UDel UD · Department of Mathematical Sciences

Shuxing Li

Doctor of Philosophy

## About

45

Publications

6,009

Reads

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923

Citations

Introduction

My research focuses on discrete structures which play fundamental roles in various mathematical branches including
combinatorics, geometry, and the mathematics of communication. Primary mathematical tools involved are character theory, algebraic number theory, and exponential sums.

Additional affiliations

November 2019 - October 2022

**Simon Fraser University**

Position

- PIMS PostDoctoral Fellow

Description

- Work with the research group led by Jonathan Jedwab

Education

September 2010 - June 2016

September 2006 - July 2010

## Publications

Publications (45)

Compressed sensing is a sampling technique which provides a fundamentally new approach to data acquisition. Comparing with traditional methods, compressed sensing makes full use of sparsity so that a sparse signal can be reconstructed from very few measurements. A central problem in compressed sensing is the construction of sensing matrices. While...

In addition to their applications in data storage, communications systems, and consumer electronics, LCD codes -- a class of linear codes -- have been employed in cryptography recently. LCD cyclic codes were referred to as reversible cyclic codes in the literature. The objective of this paper is to construct several families of reversible cyclic co...

Due to wide applications of BCH codes, the determination of their minimum distance is of great interest. However, this is a very challenging problem for which few theoretical results have been reported in the last four decades. Even for the narrow-sense primitive BCH codes, which form the most well-studied subclass of BCH codes, there are very few...

Inspired by an experimental study of energy-minimizing periodic configurations in Euclidean space, Cohn, Kumar and Sch\"urmann proposed the concept of formal duality between a pair of periodic configurations, which indicates an unexpected symmetry possessed by the energy-minimizing periodic configurations. Later on, Cohn, Kumar, Reiher and Sch\"urm...

A packing of partial difference sets is a collection of disjoint partial difference sets in a finite group $G$. This configuration has received considerable attention in design theory, finite geometry, coding theory, and graph theory over many years, although often only implicitly. We consider packings of certain Latin square type partial differenc...

A Hadamard matrix is balanced splittable if some subset of its rows has the property that the dot product of every two distinct columns takes at most two values. This definition was introduced by Kharaghani and Suda in 2019, although equivalent formulations have been previously studied using different terminology. We collate previous results phrase...

The combination of the group ring setting with the methods of character theory allows an elegant and powerful analysis of various combinatorial structures, via their character sums. These combinatorial structures include difference sets, relative difference sets, partial difference sets, hyperplanes, spreads, and LP-packings. However, the literatur...

An (n, k)-perfect sequence covering array with multiplicity \(\lambda \), denoted \(\mathrm{{PSCA}}(n,k,\lambda )\), is a multiset whose elements are permutations of the sequence \((1,2, \dots , n)\) and which collectively contain each ordered length k subsequence exactly \(\lambda \) times. The primary objective is to determine for each pair (n, k...

A Hadamard matrix is balanced splittable if some subset of its rows has the property that the dot product of every two distinct columns takes at most two values. This definition was introduced by Kharaghani and Suda in 2019, although equivalent formulations have been previously studied using different terminology. We collate previous results phrase...

An $(n,k)$-perfect sequence covering array with multiplicity $\lambda$, denoted PSCA$(n,k,\lambda)$, is a multiset whose elements are permutations of the sequence $(1,2, \dots, n)$ and which collectively contain each ordered length $k$ subsequence exactly $\lambda$ times. The primary objective is to determine for each pair $(n,k)$ the smallest valu...

The intersection distribution of a polynomial $f$ over finite field $\mathbb{F}_q$ was recently proposed by Li and Pott [\emph{Finite Fields and Their Applications, 66 (2020)}], which concerns the collective behaviour of a collection of polynomials $\{f(x)+cx \mid c \in\mathbb{F}_q\}$. The intersection distribution has an underlying geometric inter...

Let v be a product of at most three not necessarily distinct primes. We prove that there exists no strong external difference family with more than two subsets in an abelian group G of order v, except possibly when G=Cp3 and p is a prime greater than 3×1012.

A packing of partial difference sets is a collection of disjoint partial difference sets in a finite group $G$. This configuration has received considerable attention in design theory, finite geometry, coding theory, and graph theory over many years, although often only implicitly. We consider packings of certain Latin square type partial differenc...

In this paper, we propose the concepts of intersection distribution and non-hitting index, which can be viewed from two related perspectives. The first one concerns a point set S of size q+1 in the classical projective plane PG(2,q), where the intersection distribution of S indicates the intersection pattern between S and the lines in PG(2,q). The...

For a function
$f$
from
$\mathbb {F}_{2}^{n}$
to
$\mathbb {F}_{2}^{n}$
, the planarity of
$f$
is usually measured by its differential uniformity and differential spectrum. In this paper, we propose the concept of vanishing flats, which supplies a combinatorial viewpoint on the planarity. First, the number of vanishing flats of
$f$
can be...

For a function $f$ from $\mathbb{F}_2^n$ to $\mathbb{F}_2^n$, the planarity of $f$ is usually measured by its differential uniformity and differential spectrum. In this paper, we propose the concept of vanishing flats, which supplies a combinatorial viewpoint on the planarity. First, the number of vanishing flats of $f$ can be regarded as a measure...

The concept of formal duality was proposed by Cohn, Kumar and Schürmann, which reflects a remarkable symmetry among energy-minimizing periodic configurations. This formal duality was later translated into a purely combinatorial property by Cohn, Kumar, Reiher and Schürmann, where the corresponding combinatorial objects were called formally dual pai...

The intersection distribution of a polynomial $f$ over finite field $\mathbb{F}_q$ was recently proposed in Li and Pott (arXiv:2003.06678v1), which concerns the collective behaviour of a collection of polynomials $\{f(x)+cx \mid c \in \mathbb{F}_q\}$. The intersection distribution has an underlying geometric interpretation, which indicates the inte...

We propose the concepts of intersection distribution and non-hitting index, which can be viewed from two related perspectives. The first one concerns a point set $S$ of size $q+1$ in the classical projective plane $PG(2,q)$, where the intersection distribution of $S$ indicates the intersection pattern between $S$ and the lines in $PG(2,q)$. The sec...

Let $v$ be a product of at most three not necessarily distinct primes. We prove that there exists no strong external difference family with more than two subsets in abelian group $G$ of order $v$, except possibly when $G=C_p^3$ and $p$ is a prime greater than $3 \times 10^{12}$.

The weight distribution of second order q-ary Reed–Muller codes have been determined by Sloane and Berlekamp (IEEE Trans. Inform. Theory, vol. IT-16, 1970) for \(q=2\) and by McEliece (JPL Space Progr Summ 3:28–33, 1969) for general prime power q. Unfortunately, there were some mistakes in the computation of the latter one. This paper aims to provi...

The concept of formal duality was proposed by Cohn, Kumar and Schürmann, which reflects a remarkable symmetry among energy‐minimizing periodic configurations. This formal duality was later translated into a purely combinatorial property by Cohn, Kumar, Reiher and Schürmann, where the corresponding combinatorial objects were called formally dual pai...

The concept of formal duality was proposed by Cohn, Kumar and Sch\"urmann, which reflects a remarkable symmetry among energy-minimizing periodic configurations. This formal duality was later translated into a purely combinatorial property by Cohn, Kumar, Reiher and Sch\"urmann, where the corresponding combinatorial objects were called formally dual...

The weight distribution of second order $q$-ary Reed-Muller codes have been determined by Sloane and Berlekamp (IEEE Trans. Inform. Theory, vol. IT-16, 1970) for $q=2$ and by McEliece (JPL Space Programs Summary, vol. 3, 1969) for general prime power $q$. Unfortunately, there were some mistakes in the computation of the latter one. This paper aims...

Strong external difference families (SEDFs) were introduced by Paterson and Stinson as a more restrictive version of external difference families. SEDFs can be used to produce optimal strong algebraic manipulation detection codes. We characterize the parameters $(v, m, k, \lambda)$ of a nontrivial SEDF that is near-complete (satisfying $v=km+1$). W...

The concept of formal duality was proposed by Cohn, Kumar and Sch\"urmann, which reflects an unexpected symmetry among energy-minimizing periodic configurations. This formal duality was later on translated into a purely combinatorial property by Cohn, Kumar, Reiher and Sch\"urmann, where the corresponding combinatorial object was called formally du...

Symbol-pair code is a new coding framework which is proposed to correct errors in the symbol-pair read channel. In particular, maximum distance separable (MDS) symbol-pair codes are a kind of symbol-pair codes with the best possible error-correction capability. Employing cyclic and constacyclic codes, we construct three new classes of MDS symbol-pa...

Cyclic codes are widely employed in communication systems, storage devices and consumer electronics, as they have efficient encoding and decoding algorithms. BCH codes, as a special subclass of cyclic codes, are in most cases among the best cyclic codes. A subclass of good BCH codes are the narrow-sense BCH codes over GF(q) with length n = (qm–1)=(...

A linking system of difference sets is a collection of mutually related group difference sets, whose advantageous properties have been used to extend classical constructions of systems of linked symmetric designs. The central problems are to determine which groups contain a linking system of difference sets, and how large such a system can be. All...

Historically, LCD cyclic codes were referred to as reversible cyclic codes, which had applications in data storage. Due to a newly discovered application in cryptography, there has been renewed interest in LCD codes. In this paper, we explore two special families of LCD cyclic codes, which are both BCH codes. The dimensions and the minimum distance...

The generalized Hamming weights (GHWs) are fundamental parameters of linear codes. GHWs are of great interest in many applications since they convey detailed information of linear codes. In this paper, we continue the work of [10] to study the GHWs of a family of cyclic codes with arbitrary number of nonzeroes. The weight hierarchy is determined by...

Partitioned difference families are an interesting class of discrete structures which can be used to derive optimal constant composition codes. There have been intensive researches on the construction of partitioned difference families. In this paper, we consider the combinatorial approach. We introduce a new combinatorial configuration named parti...

Cyclic codes are an interesting class of linear codes due to their efficient encoding and decoding algorithms as well as their theoretical importance. BCH codes form a subclass of cyclic codes and are very important in both theory and practice as they have good error-correcting capability and are widely used in communication systems, storage device...

Constacyclic codes which generalize the classical cyclic codes have played important roles in recent constructions of many new quantum maximum distance separable (MDS) codes. However, the mathematical mechanism may not have been fully understood. In this paper, we use pseudo-cyclic codes, which is a further generalization of constacyclic codes, to...

Cyclic codes over finite fields are widely employed in communication systems, storage devices and consumer electronics, as they have efficient encoding and decoding algorithms. BCH codes, as a special subclass of cyclic codes, are in most cases among the best cyclic codes. A subclass of good BCH codes are the narrow-sense BCH codes over $\gf(q)$ wi...

The construction of group divisible designs (GDDs) is a basic problem in design theory. While there have been some methods concerning the constructions of uniform GDDs, the construction of nonuniform GDDs remains a challenging problem. In this paper, we present a new approach to the construction of nonuniform GDDs with group type and block size k....

The generalized Hamming weights (GHWs) of linear codes are fundamental
parameters, the knowledge of which is of great interest in many applications.
However, to determine the GHWs of linear codes is difficult in general. In this
paper, we study the GHWs for a family of reducible cyclic codes and obtain the
complete weight hierarchy in several cases...

Compressed sensing is a novel sampling technique that provides a fundamentally new approach to data acquisition. Comparing with the traditional method, compressed sensing asserts that a sparse signal can be reconstructed from very few measurements. A central problem in compressed sensing is the construction of sensing matrices. While random sensing...

Compressed sensing is a novel sampling theory, which provides a fundamentally new approach to data acquisition. It asserts that a sparse or compressible signal can be reconstructed from much fewer measurements than traditional methods. A central problem in compressed sensing is the construction of the sensing matrix. While random sensing matrices h...

Recently, there has been intensive research on the weight distributions of
cyclic codes. In this paper, we compute the weight distributions of three
classes of cyclic codes with Niho exponents. More specifically, we obtain two
classes of binary three-weight and four-weight cyclic codes and a class of
nonbinary four-weight cyclic codes. The weight d...

The determination of the cross correlation between an $m$-sequence and its
decimated sequence has been a long-standing research problem. Considering a
ternary $m$-sequence of period $3^{3r}-1$, we determine the cross correlation
distribution for decimations $d=3^{r}+2$ and $d=3^{2r}+2$, where $\gcd(r,3)=1$.
Meanwhile, for a binary $m$-sequence of p...

Planar functions in odd characteristic were introduced by Dembowski and
Ostrom in order to construct finite projective planes in 1968. They were also
used in the constructions of DES-like iterated ciphers, error-correcting codes,
and signal sets. Recently, a new definition of planar functions in even
characteristic was proposed by Zhou. These new p...

The known families of difference sets can be subdivided into three classes:
difference sets with Singer parameters, cyclotomic difference sets, and
difference sets with gcd$(v,n)>1$. It is remarkable that all the known
difference sets with gcd$(v,n)>1$ have the so-called character divisibility
property. In 1997, Jungnickel and Schmidt posed the pro...

The determination of weight distribution of cyclic codes involves evaluation
of Gauss sums and exponential sums. Despite of some cases where a neat
expression is available, the computation is generally rather complicated. In
this note, we determine the weight distribution of a class of reducible cyclic
codes whose dual codes may have arbitrarily ma...