arXiv subjects

Yanxun Chang

Publications and source records attributed to Yanxun Chang.

At least 19 recordsLinked to original sources

arXiv2026

Parallel covering a rhombus with equilateral triangles

Suppose that ${R}^{\alpha}$ is a rhombus with side length $1$ and with an interior angle $\alpha$, where $0<\alpha\leq \frac{\pi}{2}$. Let $\triangle$ be an equilateral triangle with a side parallel to a side of ${R}^{\alpha}$ and let $\{\triangle_{n}\}$ be a collection of homothetic copies of $\triangle$. In this paper, we show the following two results: if $0<\alpha\leq\frac{\pi}{3}$ and the sum of the areas of equilateral triangles from $\{\triangle_{n}\}$ is at least $\frac{\sqrt{3}}{4}(1+\cos\alpha+\frac{\sqrt{3}}{3}\sin\alpha)^{2}$, then these equilateral triangles can parallel cover the rhombus ${R}^{\alpha}$; if $\frac{\pi}{3}<\alpha\leq\frac{\pi}{2}$ and the sum of the areas of equilateral triangles from $\{\triangle_{n}\}$ is at least $\frac{\sqrt{3}}{4}(1+\frac{2\sqrt{3}}{3}\sin\alpha)^{2}$, then they can parallel cover the rhombus ${R}^{\alpha}$. Furthermore, these bounds are optimal on their respective intervals.

math.MG↗

arXiv2026

Constructions of two-dimensional optical orthogonal codes of weight three

The study of optical orthogonal codes has been motivated by an application in an optical code-division multiple access system. This paper focuses on optimal two-dimensional optical orthogonal codes with autocorrelation and cross-correlation both equal to $1$. By examining the structures of $n$-cyclic group divisible packings and semi-cyclic incomplete holey group divisible designs, we present new combinatorial constructions for two-dimensional $(m\times n,k,1)$-optical orthogonal codes. As a consequence, the exact number of codewords of an optimal two-dimensional $(m\times n,3,1)$-optical orthogonal code is determined for any positive integers $m$ and $n$.

math.CO↗

arXiv2026

Cardinality-consistent flag codes with longer type vectors

Flag codes generalize constant dimension codes by considering sequences of nested subspaces with prescribed dimensions as codewords. A comprehensive construction, which unites cyclic orbit flag codes, yields two families of flag codes on $\mathbb{F}^n_q$ (where $n=sk+h$ with $s\geq 2$ and $0\leq h < k$): optimum distance flag codes of the longest possible type vector $(1, 2, \ldots, k, n-k, \ldots, n-1)$ and flag codes with longer type vectors $(1, 2, \ldots, k+h, 2k+h, \ldots, (s-2)k+h, n-k, \ldots, n-1)$. These flag codes achieve the same cardinality $\sum^{s-1}_{i=1}q^{ik+h}+1$.

cs.IT↗

arXiv2025

The existence of pyramidal Steiner triple systems over abelian groups

A Steiner triple system STS$(v)$ is called $f$-pyramidal if it has an automorphism group fixing $f$ points and acting sharply transitively on the remaining $v-f$ points. In this paper, we focus on the STSs that are $f$-pyramidal over some abelian group. Their existence has been settled only for the smallest admissible values of $f$, that is, $f=0,1,3$. In this paper, we complete this result and determine, for every $f>3$, the spectrum of values $(f,v)$ for which there is an $f$-pyramidal STS$(v)$ over an abelian group. This result is obtained by constructing difference families relative to a suitable partial spread.

math.CO↗

arXiv2024

The weight distributions of some linear codes derived from Kloosterman sums

Linear codes with few weights have applications in data storage systems, secret sharing schemes, and authentication codes. In this paper, some kinds of p-ary linear codes with few weights are constructed by use of the given de ning set, where p is a prime. Their weight distributions are determined based on Kloosterman sums over nite elds. In addition, some linear codes we given is minimal.

cs.IT↗

arXiv2023

Partitionable sets, almost partitionable sets and their applications

This paper introduces almost partitionable sets to generalize the known concept of partitionable sets. These notions provide a unified frame to construct $\mathbb{Z}$-cyclic patterned starter whist tournaments and cyclic balanced sampling plans excluding contiguous units. The existences of partitionable sets and almost partitionable sets are investigated. As an application, a large number of optical orthogonal codes achieving the Johnson bound or the Johnson bound minus one are constructed.

math.CO↗

arXiv2023

Cyclic relative difference families with block size four and their applications

Given a subgroup $H$ of a group $(G,+)$, a $(G,H,k,1)$ difference family (DF) is a set $\mathcal F$ of $k$-subsets of $G$ such that $\{f-f':f,f'\in F, f\neq f',F\in \mathcal F\}=G\setminus H$. Let $g\mathbb Z_{gh}$ is the subgroup of order $h$ in $\mathbb Z_{gh}$ generated by $g$. A $(\mathbb Z_{gh},g\mathbb Z_{gh},k,1)$-DF is called cyclic and written as a $(gh,h,k,1)$-CDF. This paper shows that for $h\in\{2,3,6\}$, there exists a $(gh,h,4,1)$-CDF if and only if $gh\equiv h\pmod{12}$, $g\geq 4$ and $(g,h)\not\in\{(9,3),(5,6)\}$. As a corollary, it is shown that a 1-rotational S$(2,4,v)$ exists if and only if $v\equiv4\pmod{12}$ and $v\neq 28$. This solves the long-standing open problem on the existence of a 1-rotational S$(2,4,v)$. As another corollary, we establish the existence of an optimal $(v,4,1)$-optical orthogonal code with $\lfloor(v-1)/12\rfloor$ codewords for any positive integer $v\equiv 1,2,3,4,6\pmod{12}$ and $v\neq 25$. We also give applications of our results to cyclic group divisible designs with block size four and optimal cyclic $3$-ary constant-weight codes with weight four and minimum distance six.

math.CO↗

arXiv2022

Several problems on reduced spherical polygons of thickness less than π/2

The present paper aims to solve some problems proposed by Lassak about the reduced spherical polygons. The main result is to show that the regular spherical n-gon has the minimal perimeter among all reduced spherical polygons of fixed thickness less than π/2 and with at most n vertices. In addition, we determine the maximal diameter of every reduced spherical polygons with a fixed thickness less than π/2. We also find the smallest spherical radius that contains every reduced spherical polygons with a fixed thickness less than π/2.

math.MG↗

arXiv2020

The area of reduced spherical polygons

We confirm two conjectures of Lassak on the area of reduced spherical polygons. The area of every reduced spherical non-regular $n$-gon is less than that of the regular spherical $n$-gon of the same thickness. Moreover, the area of every reduced spherical polygon is less than that of the regular spherical odd-gons of the same thickness and whose number of vertices tends to infinity.

math.MG↗

arXiv2020

New bounds on the maximum size of Sperner partition systems

An $(n,k)$-Sperner partition system is a collection of partitions of some $n$-set, each into $k$ nonempty classes, such that no class of any partition is a subset of a class of any other. The maximum number of partitions in an $(n,k)$-Sperner partition system is denoted $\mathrm{SP}(n,k)$. In this paper we introduce a new construction for Sperner partition systems and use it to asymptotically determine $\mathrm{SP}(n,k)$ in many cases as $\frac{n}{k}$ becomes large. We also give a slightly improved upper bound for $\mathrm{SP}(n,k)$ and exhibit an infinite family of parameter sets $(n,k)$ for which this bound is tight.

math.CO↗

arXiv2019

Parallel multilevel constructions for constant dimension codes

Constant dimension codes (CDCs), as special subspace codes, have received a lot of attention due to their application in random network coding. This paper introduces a family of new codes, called rank metric codes with given ranks (GRMCs), to generalize the parallel construction in [Xu and Chen, IEEE Trans. Inf. Theory, 64 (2018), 6315--6319] and the classic multilevel construction. A Singleton-like upper bound and a lower bound for GRMCs derived from Gabidulin codes are given. Via GRMCs, two effective constructions for CDCs are presented by combining the parallel construction and the multilevel construction. Many CDCs with larger size than the previously best known codes are given. The ratio between the new lower bound and the known upper bound for $(4δ,2δ,2δ)_q$-CDCs is calculated. It is greater than 0.99926 for any prime power $q$ and any $δ\geq 3$.

math.CO↗

arXiv2019

Triangle decompositions of $λK_v-λK_w-λK_u$

Denote by $λK_v$ the complete graph of order $v$ with multiplicity $λ$. Let $λK_v-λK_w-λK_u$ be the graph obtained from $λK_v$ by the removal of the edges of two vertex disjoint complete multi-subgraphs with multiplicity $ λ$ of orders $ w $ and $ u $, respectively. When $λ$ is odd, it is shown that there exists a triangle decomposition of $λK_v-λK_w-λK_u$ if and only if $v\geq w+u+\max\{u,w\}$, $ λ\left({v\choose 2}-{u\choose 2}-{w\choose 2}\right) \equiv 0 \pmod 3$ and $λ(v-w) \equiv λ(v-u) \equiv λ(v-1) \equiv 0 \pmod 2$. When $λ$ is even, it is shown that for large enough $v$, the elementary necessary conditions for the existence of a triangle decomposition of $λK_v-λK_w-λK_u$ are also sufficient.

math.CO↗

arXiv2019

Several classes of optimal Ferrers diagram rank-metric codes

Four constructions for Ferrers diagram rank-metric (FDRM) codes are presented. The first one makes use of a characterization on generator matrices of a class of systematic maximum rank distance codes. By introducing restricted Gabidulin codes, the second construction is presented, which unifies many known constructions for FDRM codes. The third and fourth constructions are based on two different ways to represent elements of a finite field $\mathbb F_{q^m}$ (vector representation and matrix representation). Each of these constructions produces optimal codes with different diagrams and parameters.

math.CO↗

arXiv2019

Strong difference families of special types

Strong difference families of special types are introduced to produce new relative difference families from the point of view of both asymptotic existences and concrete examples. As applications, group divisible designs of type $30^u$ with block size $6$ are discussed, $r$-rotational balanced incomplete block designs with block size $6$ are derived for $r\in\{6,10\}$, and several classes of optimal optical orthogonal codes with weight $5$, $6$, $7$, or $8$ are obtained.

math.CO↗

arXiv2019

Leaves for packings with block size four

We consider maximum packings of edge-disjoint $4$-cliques in the complete graph $K_n$. When $n \equiv 1$ or $4 \pmod{12}$, these are simply block designs. In other congruence classes, there are necessarily uncovered edges; we examine the possible `leave' graphs induced by those edges. We give particular emphasis to the case $n \equiv 0$ or $3 \pmod{12}$, when the leave is $2$-regular. Colbourn and Ling settled the case of Hamiltonian leaves in this case. We extend their construction and use several additional direct and recursive constructions to realize a variety of $2$-regular leaves. For various subsets $S \subseteq \{3,4,5,\dots\}$, we establish explicit lower bounds on $n$ to guarantee the existence of maximum packings with any possible leave whose cycle lengths belong to $S$.

math.CO↗

arXiv2019

Constructions for optimal Ferrers diagram rank-metric codes

Optimal rank-metric codes in Ferrers diagrams can be used to construct good subspace codes. Such codes consist of matrices having zeros at certain fixed positions. This paper generalizes the known constructions for Ferrers diagram rank-metric (FDRM) codes. Via a criteria for linear maximum rank distance (MRD) codes, an explicit construction for a class of systematic MRD codes is presented, which is used to produce new optimal FDRM codes. By exploring subcodes of Gabidulin codes, if each of the rightmost $δ-1$ columns in Ferrers diagram $\cal F$ has at least $n-r$ dots, where $r$ is taken in a range, then the conditions that an FDRM code in $\cal F$ is optimal are established. The known combining constructions for FDRM code are generalized by introducing the concept of proper combinations of Ferrers diagrams.

math.CO↗

arXiv2016

Determination of sizes of optimal three-dimensional optical orthogonal codes of weight three with the AM-OPP restriction

In this paper, we further investigate the constructions on three-dimensional $(u\times v\times w,k,1)$ optical orthogonal codes with the at most one optical pulse per wavelength/time plane restriction (briefly AM-OPP $3$-D $(u\times v\times w,k,1)$-OOCs) by way of the corresponding designs. Several new auxiliary designs such as incomplete holey group divisible designs and incomplete group divisible packings are introduced and therefore new constructions are presented. As a consequence, the exact number of codewords of an optimal AM-OPP $3$-D $(u\times v\times w,3,1)$-OOC is finally determined for any positive integers $v,w$ and $u\geq3$.

cs.IT↗

arXiv2014

Enumerations of (K_4-e)-designs with small orders

It is established that up to isomorphism,there are only one (K_4-e)-design of order 6, three (K_4-e)-designs of order 10 and two (K_4-e)-designs of order 11. As an application of our enumerative results, we discuss the fine triangle intersection problem for (K_4-e)-designs of orders v=6,10,11.

math.CO↗