On Skew Hadamard difference sets

Non-symmetric class $2$ association schemes obtained by doubling of skew-Hadamard matrices are non-schurian

We can obtain a non-symmetric class $2$ association scheme by a skew-Hadamard matrix. We begin with a skew-Hadamard matrix of order $n$, construct a skew-Hadamard matrix of order $2n$ by doubling construction, and a non-symmetric class $2$ association scheme of order $2n-1$. We will show that the association scheme obtained in this way never be schurian if $n$ is greater than or equal to $8$.

Skew Hadamard difference sets from cyclotomic strongly regular graphs

We find new constructions of infinite families of skew Hadamard difference sets in elementary abelian groups under the assumption of the existence of cyclotomic strongly regular graphs. Our construction is based on choosing cyclotomic classes in finite fields.

Hadamard 矩阵：276、292 阶斜 Hadamard 矩阵与 372 阶对称 Hadamard 矩阵

Hadamard matrices: skew of orders 276, 292 and symmetric 372

本文针对最小未知阶数 v=69 的斜 Hadamard 矩阵存在性问题，给出多种构造方法，并首次构造出 292 阶斜 Hadamard 矩阵和 372 阶对称 Hadamard 矩阵。

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 po…

Skew-Hadamard matrices of orders 436, 580 and 988 exist

We construct two difference families on each of the cyclic groups of order 109, 145 and 247, and use them to construct skew-Hadamard matrices of orders 436, 580 and 988. Such difference families and matrices are constructed here for the first time. The matrices are constructed by using the Goethals-Seidel array.

Morphisms of Skew Hadamard Matrices

Quaternary unit Hadamard (QUH) matrices were introduced by Fender, Kharagani and Suda along with a method to construct them at prime power orders. We present a novel construction of real Hadamard matrices from QUH matrices. Our construction recovers the result by Mukhopadhyay on the existence of real Hadamard matrices of order $q^n+q^{n-1}$ for each prime power $q\equiv 3\mod 4$. Furthermore we provide nonexistence conditions for QUH matrices.

Cyclotomic Constructions of Skew Hadamard Difference Sets

We revisit the old idea of constructing difference sets from cyclotomic classes. Two constructions of skew Hadamard difference sets are given in the additive groups of finite fields using unions of cyclotomic classes of order $N=2p_1^m$, where $p_1$ is a prime and $m$ a positive integer. Our main tools are index 2 Gauss sums, instead of cyclotomic numbers.

Skew-Hadamard matrices of orders 188 and 388 exist

We construct several difference families on cyclic groups of orders 47 and 97, and use them to construct skew-Hadamard matrices of orders 188 and 388. Such difference families and matrices are constructed here for the first time. The matrices are constructed by using the Goethals-Seidel array.

一类斜正则四元Hadamard矩阵

A class of skew-regular quaternary Hadamard matrices

本文研究了一类新的斜正则四元Hadamard矩阵，针对每个奇素数幂p，证明了阶为1+p²的矩阵存在性，其行和恒为1-pi，为生成具有极大超额的Hadamard矩阵提供了稳健框架。

Constructions of Strongly Regular Cayley Graphs and Skew Hadamard Difference Sets from Cyclotomic Classes

In this paper, we give a construction of strongly regular Cayley graphs and a construction of skew Hadamard difference sets. Both constructions are based on choosing cyclotomic classes in finite fields, and they generalize the constructions given by Feng and Xiang \cite{FX111,FX113}. Three infinite families of strongly regular graphs with new parameters are obtained. The main tools that we employed are index 2 Gauss sums, instead of cyclotomic numbers.

Skew Hadamard Difference Sets from Dickson Polynomials of Order 7

Skew Hadamard difference sets are an interesting topic of study for over seventy years. For a long time, it had been conjectured the classical Paley difference sets (the set of nonzero quadratic residues in $\mathbb{F}_q$ where $q \equiv 3 \bmod{4}$) were the only example in abelian groups. In 2006, the first author and Yuan disproved this conjecture by showing that the image set of $\mathcal{D}_5(x^2,u)$ is a new skew Hadamard difference set in $(\mathbb{F}_{3^m},+)$ with $m$ odd, where $\mathcal{D}_n(x,u)$ denotes the first kind of Dickson polynomials of order $n$ and $u \in \mathbb{F}_q^*$.…

Pure Gauss sums and skew Hadamard difference sets

Chowla~(1962), McEliece~(1974), Evans~(1977, 1981) and Aoki~(1997, 2004, 2012) studied Gauss sums, some integral powers of which are in the field of rational numbers. Such Gauss sums are called {\it pure}. In particular, Aoki (2004) gave a necessary and sufficient condition for a Gauss sum to be pure in terms of Dirichlet characters modulo the order of the multiplicative character involved. In this paper, we study pure Gauss sums with odd extension degree $f$ and classify them for $f=5,7,9,11,13,17,19,23$ based on Aoki's theorem. Furthermore, we characterize a special subclass of pure Gauss su…

On Construction of Hadamard Matrices

In this article, a series of Hadamard matrix has been developed using some block matrices with the help of skew Hadamard matrix. Basically an internal structure of skew Hadamard matrix has been changed with some block matrices using kronecker product. For some parameter, Hadamard matrices of order 4t where t is an integer, has been found.

通过循环根联合构造一个阶数为1252的显式斜哈达玛矩阵

An Explicit Skew-Hadamard Matrix of Order 1252 via Cyclotomic Unions

本文通过循环根联合构造了一个阶数为1252的显式斜哈达玛矩阵，并验证了其结构和代数性质。

Difference families, skew Hadamard matrices, and Critical groups of doubly regular tournaments

In this paper we investigate the structure of the critical groups of doubly regular tournaments (DRTs) associated with skew Hadamard difference families (SDFs) with one, two, or four blocks. Brown and Reid found the existence of a skew Hadamard matrix of order $n+1$ to be equivalent to the existence of a DRT on $n$ vertices. A well known construction of a skew Hadamard matrix order $n$ is by constructing skew Hadamard difference sets in abelian groups of order $n-1$. The Paley skew Hadamard matrix is an example of one such construction. Szekeres and Whiteman constructed skew Hadamard matrices …

实辛空间中的等角紧框架：Zauner猜想与斜Hadamard猜想

Equiangular tight frames in real symplectic space: Zauner's conjecture and the skew Hadamard conjecture

本文引入实辛空间中的等角紧框架概念，提出关于其存在性的猜想，并证明“辛Zauner猜想”等价于斜Hadamard猜想，其证明涉及对竞赛图中“菱形”子图的计数。

Strongly Regular Cayley Graphs, Skew Hadamard Difference Sets, and Rationality of Relative Gauss Sums

In this paper, we give constructions of strongly regular Cayley graphs and skew Hadamard difference sets. Both constructions are based on choosing cyclotomic classes in finite fields, and our results generalize ten of the eleven sporadic examples of cyclotomic strongly regular graphs given by Schmidt and White [24] and several of subfield examples into infinite families. These infinite families of strongly regular graphs have new parameters. The main tools that we employed are relative Gauss sums instead of explicit evaluations of Gauss sums.

论某些非交换群中不存在斜阿达马差集

On the non-existence of skew-Hadamard difference sets in certain non-abelian groups

本文通过有理群代数结构，证明了若幂零群存在斜阿达马差集，则该群必为p-群，首次给出非交换群中斜阿达马差集的一般结构限制。

一个 Hadamard 矩阵构造数据库

A database of constructions of Hadamard matrices

本文在 SageMath 中构建覆盖阶≤1208的 Hadamard 与斜 Hadamard 矩阵构造数据库，修正文献错误，并更新至阶2999/999的最小指数表，还揭示其与 Riesel 数的关系。

Supplementary difference sets with symmetry for Hadamard matrices

First we give an overview of the known supplementary difference sets (SDS) (A_i), i=1..4, with parameters (n;k_i;d), where k_i=|A_i| and each A_i is either symmetric or skew and k_1 + ... + k_4 = n + d. Five new Williamson matrices over the elementary abelian groups of order 25, 27 and 49 are constructed. New examples of skew Hadamard matrices of order 4n for n=47,61,127 are presented. The last of these is obtained from a (127,57,76)-difference family that we have constructed. An old non-published example of G-matrices of order 37 is also included.

(36,15,6) Hadamard差集的扩散构造

Spread Construction for (36,15,6) Hadamard Difference Sets

本文提出一种直接的扩散构造，解释了九个群中32个不等价的(36,15,6)差集，并通过该构造的变体生成了第九个群中的三个差集。

Generalized constructions of Menon-Hadamard difference sets

We revisit the problem of constructing Menon-Hadamard difference sets. In 1997, Wilson and Xiang gave a general framework for constructing Menon-Hadamard difference sets by using a combination of a spread and four projective sets of type Q in ${\mathrm{PG}}(3,q)$. They also found examples of suitable spreads and projective sets of type Q for $q=5,13,17$. Subsequently, Chen (1997) succeeded in finding a spread and four projective sets of type Q in ${\mathrm{PG}}(3,q)$ satisfying the conditions in the Wilson-Xiang construction for all odd prime powers $q$. Thus, he showed that there exists a Men…

A class of cyclic $(v;k_1,k_2,k_3;λ)$ difference families with $v \equiv 3 \pmod{4}$ a prime

We construct several cyclic $(v;k_1,k_2,k_3;λ)$ difference families with $v\equiv3 \pmod{4}$ a prime and $λ=k_1+k_2+k_3-(3v-1)/4$. Such families can be used in conjunction with the well-known Paley-Todd difference sets to construct skew-Hadamard matrices of order $4v$. Our main result is that we have constructed for the first time the example of skew-Hadamard matrices of orders $4\cdot239=956$ and $4\cdot331=1324$.

Symmetric Bush-type Hadamard matrices of order $4m^4$ exist for all odd $m$

Using reversible Hadamard difference sets, we construct symmetric Bush-type Hadamard matrices of order $4m^4$ for all odd integer $m$.

Schur Ring over Group $\Z_{2}^{n}$, Circulant $S-$Sets Invariant by Decimation and Hadamard Matrices

In this paper a variety of issues are discussed, Schur ring, $S$-sets, circulant orbits, decimation operator and Hadamard matrices and their relation between them is shown. Firstly we define the complete $S$-sets. Next, we study the structure of Schur ring with circulant basic sets over $\Z_{2}^{n}$ and we define the free and non-free circulant $S$-sets, the symmetric, non-symmetric and antisymmetric circulant $S$-sets. We prove that all this $S$-sets are invariants under decimation. Finally, we prove that if a Hadamard matrix exist then this is contained in a complete $S$-set. Also, we prove …

具有对称或斜基块的 Goethals–Seidel 差族

Goethals--Seidel difference families with symmetric or skew base blocks

本文研究循环群中基块为对称或斜的 Goethals–Seidel 差族，通过扩展已有计算完成奇数 v<50 的分类，并构造多个新矩阵实例。

Some results on minimum skew zero forcing sets, and skew zero forcing number

Let $G$ be a graph, and $Z$ a subset of its vertices, which we color black, while the remaining are colored white. We define the skew color change rule as follows: if $u$ is a vertex of $G$, and exactly one of its neighbors $v$, is white, then change the color of $v$ to black. A set $Z$ is a skew zero forcing set for $G$ if the application of the skew color change rule (as many times as necessary) will result in all the vertices in $G$ colored black. A set $Z$ is a minimum skew zero forcing set for $G$ if it is a skew zero forcing set for $G$ of least cardinality. The skew zero forcing number …

Kochen-Specker sets and Hadamard matrices

We introduce a new class of complex Hadamard matrices which have not been studied previously. We use these matrices to construct a new infinite family of parity proofs of the Kochen-Specker theorem. We show that the recently discovered simple parity proof of the Kochen-Specker theorem is the initial member of this infinite family.

一类高斯型Hadamard差集

A Family of Gauss Type Hadamard Difference Sets

本文构造了一类新的高斯型Hadamard差集，通过特定群上的构建集和递归方法，证明了非朴素特征值存在的充要条件。

On properties of Karlsson Hadamards and sets of Mutually Unbiased Bases in dimension six

The complete classification of all 6x6 complex Hadamard matrices is an open problem. The 3-parameter Karlsson family encapsulates all Hadamards that have been parametrised explicitly. We prove that such matrices satisfy a non-trivial constraint conjectured to hold for (almost) all 6x6 Hadamard matrices. Our result imposes additional conditions in the linear programming approach to the mutually unbiased bases problem recently proposed by Matolcsi et al. Unfortunately running the linear programs we were unable to conclude that a complete set of mutually unbiased bases cannot be constructed from …

循环理论中的斜部分差集与部分差族

Cyclotomic skew partial difference sets and partial difference families

本文研究了循环理论在构造斜部分差集中的应用，提出了新的构造方法，并探讨了斜部分差集与部分差族之间的关系。

关于无斜角集

On skew corner-free sets

本研究构造了规模为$n^{5/4}$的二维网格无斜角集，证否了Kevin Pratt的猜想，同时给出了有限域乘积空间中无斜角集的规模上界。

最小 PSD 迫使集与最小斜迫使集的重构

Reconfiguration of Minimum PSD Forcing Sets and Minimum Skew Forcing Sets

本文研究最小 PSD 迫使集与最小斜迫使集的重构图，刻画其在令牌交换和令牌滑动规则下的结构与可实现性，并推广到常见图参数及零迫使变体。

Note on divisible difference sets from Galois rings GR(9,n)

In this note, we give a new construction of divisible difference sets in ${\Bbb Z}_{9}^n$ using Galois ring $GR(3^2,n)$ under the assumption of the existence of skew Hadamard difference sets in ${\Bbb F}_{3^n}$.

Zero loci of nullvectors and skew zero forcing in graphs and hypergraphs

There is interesting internal structure in the nullspaces of graph and hypergraph adjacency matrices, especially for trees, bipartite graphs, and related combinatorial classes. The zero loci of nullvectors, i.e., their zero coordinates' indices, encode information about matchings, coverings, and edges' influence on rank. This set system is the lattice of flats of a ``kernel matroid'', a subsystem of which are the ``stalled'' sets closed under skew zero forcing (SZF), a graph percolation/infection model known to have connections with rank and nullity. For a wide variety of graphs, the lattice o…

若干新的Hadamard分拆差族

A few more Hadamard Partitioned Difference Families

本文针对Hadamard分拆差族（HPDF）已知实例极少的问题，提出了若干群中的三个新HPDF实例，并通过简单论证初步研究了给定参数下HPDF的存在性，为递归构造更多分拆差族提供了新基础。

具有两条截线的 $\mathbb P^3$ 中相错直线上的 Geproci 集

Geproci sets on skew lines in $\mathbb P^3$ with two transversals

本文研究具有两条公共截线的相错直线上的 $[4,n]$-geproci 集，证明 $n\leq 6$ 且其均包含于既有标准构造，并提出适用于任意 $m$ 的猜想性表示。

On a family of symmetric rational functions

This paper is about a family of symmetric rational functions that form a one-parameter generalization of the classical Hall-Littlewood polynomials. We introduce two sets of (skew and non-skew) functions that are akin to P and Q Hall-Littlewood polynomials. We establish (a) a combinatorial formula that represents our functions as partition functions for certain path ensembles in the square grid; (b) symmetrization formulas for non-skew functions; (c) identities of Cauchy and Pieri type; (d) explicit formulas for principal specializations; (e) two types of orthogonality relations for non-skew fu…

阿贝尔差集中的禁止乘子

Forbidden multipliers in abelian difference sets

该研究发现固定阿贝尔群大子群的群自同构（除特定对合外）无法作为非平凡阿贝尔差集的乘子，结合乘子定理排除了部分差集存在性并推导了数值乘子群的界。

On the existence of cyclic difference sets with small parameters

Previous surveys by Baumert and Lopez and Sanchez have resolved the existence of cyclic (v,k,lambda) difference sets with k <= 150, except for six open cases. In this paper we show that four of those difference sets do not exist. We also look at the existence of difference sets with k <= 300 and cyclic Hadamard difference sets with v <= 10,000. Finally, we extend an earlier search of the second author to show that no cyclic projective planes exist with non-prime power orders up to two billion.

Complex Hadamard matrices and the Spectral Set Conjecture

By analyzing the connection between complex Hadamard matrices and spectral sets we prove the direction ``spectral -> tile'' of the Sectral Set Conjecture for all sets A of size at most 5 in any finite Abelian group. This result is then extended to the infinite grid $\Z^d$ for any dimension d, and finally to Euclidean space. It was pointed out recently by Tao that the corresponding statement fails for |A|=6 in the group $\Z_3^5$, and this observation quickly led to the failure of the Spectral Set Conjecture in $\R^5$ (Tao), and subsequently in $\R^4$ (Matolcsi). In the second part of this note …

斜无角集的改进界

Improved bounds for skew corner-free sets

本文构造了大小为 $n^2\exp(-O(\sqrt{\log n}))$ 的斜无角子集以改进下界，并证明了任何此类集合大小至多为 $O(n^2(\log n)^{-c})$ 以改进上界。

无斜角集合的强界

Strong Bounds for Skew-Corner-Free Sets

本文研究网格中无斜角集合的最大规模，通过推广 Kelley–Meka 的三项等差数列方法，证明其大小上界接近 Beker 的已知下界。

不含斜角的集合的良好界

Good bounds for sets lacking skew corners

针对Pratt提出的不含非平凡斜角的二维网格集合最大规模问题，研究者提出Kelley-Meka方法的二维变体，证明了带绝对常数的上界，结合已有构造本质解决该问题，另有团队独立得到相似结果。

由二元和三元数字集生成的莫兰型测度的谱性

Spectrality of Moran-Type Measures Generated by Two-Element and Three-Element Digit Sets

研究由二元和三元数字集生成的莫兰型测度的谱性，核心方法是依据卷积中各项傅里叶变换零集的因子数量判定，贡献为给出其为谱测度的充要条件，并揭示无哈达玛三元组时问题更复杂困难。

Matrices, Characters and Descents

A new family of asymmetric matrices of Walsh-Hadamard type is introduced. We study their properties and, in particular, compute their determinants and discuss their eigenvalues. The invertibility of these matrices implies that certain character formulas are invertible, yielding expressions for the cardinalities of sets of combinatorial objects with prescribed descent sets in terms of character values of the symmetric group.