Mikhail Muzychuk, Qing Xiang
Using reversible Hadamard difference sets, we construct symmetric Bush-type Hadamard matrices of order $4m^4$ for all odd integer $m$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We define Bush-type generalized Hadamard matrices over abelian groups and construct symmetric Bush-type generalized Hadamard matrices over the additive group of finite field $\mathbb{F}_q$, $q$ a prime power. We then show and study an association scheme obtained from such generalized Hadamard matrices.
It was shown by LeCompte, Martin, and Oweans in 2010 that the existence of mutually unbiased Hadamard matrices and the identity matrix, which coincide with mutually unbiased bases, is equivalent to that of a $Q$-polynomial association scheme of class four which is both $Q$-antipodal and $Q$-bipartite. We prove that the existence of a set of mutually unbiased Bush-type Hadamard matrices is equivalent to that of an association scheme of class five. As an application of this equivalence, we obtain the upper bound of the number of mutually unbiased Bush-type Hadamard matrices of order $4n^2$ to be…
In this paper we introduce a switching for 2-designs. We illustrate this method by applying it to some symmetric (64,28,12) designs. In that way we obtain six new symmetric (64,28,12) designs. Further, we show that this type of switching can be applied to any symmetric design related to a Bush-type Hadamard matrix.
We have extended the Paley constructions for Hadamard matrices and obtained some series of Hadamard matrices. Especially Paley construction-II is applicable for odd prime power q is congruent to 1(mod 4) however our method is applicable for any odd prime power. Some of which are non-isomorphic to the matrices obtained by standard Paley constructions, Sylvester construction, Williamson construction or combination of either of these constructions with Sylvester construction. In fact we have used the conference matrix of order prime power q in some different manner which is applicable for any odd…
We continue our systematic search for symmetric Hadamard matrices based on the so called propus construction. In a previous paper this search covered the orders $4v$ with odd $v\le41$. In this paper we cover the cases $v=43,45,47,49,51$. The odd integers $v<120$ for which no symmetric Hadamard matrices of order $4v$ are known are the following: $$47,59,65,67,73,81,89,93,101,103,107,109,113,119.$$ By using the propus construction, we found several symmetric Hadamard matrices of order $4v$ for $v=47,73,113$.
We show that 138 odd values of n less than 10000 for which one knows how to construct a Hadamard matrix of order 4n have been overlooked in the recent handbook of combinatorial designs. There are four additional odd n, namely 191, 5767, 7081 and 8249, in that range for which we can construct a Hadamard matrix of order 4n. Our exhaustive computer searches show that the near-normal sequences NN(n) exist for n=36,38,40. Thus the Yang conjecture on the existence of NN(n) for all even n has been verified for n <= 40 but it still remains open.
Self-dual Hadamard bent sequences
本文研究长度至多196的自对偶Hadamard bent序列，结合穷举、线性代数与Groebner基生成实例，提出存在性猜想并给出计算强自同构群的高效算法。
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 $u \in \mathbb{F}_q^*$.…
We introduce mutually unbiased complex Hadamard (MUCH) matrices and show that the number of MUCH matrices of order 2n, n odd, is at most 2 and the bound is attained for n = 1,5,9. Furthermore, we prove that certain pairs of mutually unbiased complex Hadamard matrices of order m can be used to construct pairs of unbiased real Hadamard matrices of order 2m. As a consequence we generate a new pair of unbiased real Hadamard matrices of order 36.
We update the list of odd integers n<10000 for which an Hadamard matrix of order 4n is known to exist. We also exhibit the first example of base sequences BS(40,39). Consequently, there exist T-sequences TS(n) of length n=79. The first undecided case has the length n=97.
In this paper a method for embedding cocyclic submatrices with ``large'' determinants of orders 2t in certain cocyclic Hadamard matrices of orders 4t is described (t an odd integer). If these determinants attain the largest possible value, we are embedding D-optimal designs. Applications to the pivot values that appear when Gaussian Elimination with complete pivoting is performed on these cocyclic Hadamard matrices are studied.
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 complex Hadamard matrix of order 94
本文首次利用计算机辅助方法构造了阶数为94的复Hadamard矩阵，通过修改基本块构造方法并使用特定的循环矩阵实现。
A search for Hadamard matrices of Williamson type
本文研究了一种特殊的Williamson类型矩阵，即近Williamson矩阵，通过计算机搜索发现所有奇数阶至35的非等价矩阵，并证明所有奇数阶至63的此类矩阵存在，从而找到首个118阶四元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。
Some new symmetric Hadamard matrices
本文构造了阶数508和764的首批对称Hadamard矩阵，采用propus构造法，并验证了相关猜想对v=55,57,59,61,63成立。
\textit{Propus} (which means twins) is a construction method for orthogonal $\pm 1$ matrices based on a variation of the Williamson array called the \textit{propus array} \[ \begin{matrix*}[r] A& B & B & D B& D & -A &-B B& -A & -D & B D& -B & B &-A. \end{matrix*} \] This construction designed to find symmetric Hadamard matrices was originally based on circulant symmetric $\pm 1$ matrices, called \textit{propus matrices}. We also give another construction based on symmetric Williamson-type matrices. We give constructions to find symmetric propus-Hadamard matrices for 57 orders $4n$, $n < 200$ o…
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…
We study the existence and construction of circulant matrices $C$ of order $n\geq2$ with diagonal entries $d\geq0$, off-diagonal entries $\pm1$ and mutually orthogonal rows. These matrices generalize circulant conference ($d=0$) and circulant Hadamard ($d=1$) matrices. We demonstrate that matrices $C$ exist for every order $n$ and for $d$ chosen such that $n=2d+2$, and we find all solutions $C$ with this property. Furthermore, we prove that if $C$ is symmetric, or $n-1$ is prime, or $d$ is not an odd integer, then necessarily $n=2d+2$. Finally, we conjecture that the relation $n=2d+2$ holds fo…
A generalisation of bent vectors for Butson Hadamard matrices
本文推广了Butson哈达玛矩阵的bent向量概念，定义了自对偶和共轭自对偶bent向量，利用代数数论证明其阶条件与不存在性结果，通过张量构造等给出例子，并应用于推广Reed-Muller码的非线性码覆盖半径研究。
It is conjectured that Hadamard matrices exist for all orders $4t$ ($t>0$). However, despite a sustained effort over more than five decades, the strongest overall existence results are asymptotic results of the form: for all odd natural numbers $k$, there is a Hadamard matrix of order $k2^{[a+b\log_2k]}$, where $a$ and $b$ are fixed non-negative constants. To prove the Hadamard Conjecture, it is sufficient to show that we may take $a=2$ and $b=0$. Since Seberry's ground-breaking result, which showed that we may take $a=0$ and $b=2$, there have been several improvements where $b$ has been by st…
We classify all the cocyclic Butson Hadamard matrices $\mathrm{BH}(n,p)$ of order $n$ over the $p$th roots of unity for an odd prime $p$ and $np\leq 100$. That is, we compile a list of matrices such that any cocyclic $\mathrm{BH}(n,p)$ for these $n$, $p$ is equivalent to exactly one element in the list. Our approach encompasses non-existence results and computational machinery for Butson and generalized Hadamard matrices that are of independent interest.
Let $K$ be a finite abelian group and let $\exp(K)$ denote the least common multiple of the orders of the elements of $K$. A $BH(K,h)$ matrix is a $K$-invariant $|K|\times |K|$ matrix $H$ whose entries are complex $h$th roots of unity such that $HH^*=|K|I$, where $H^*$ denotes the complex conjugate transpose of $H$, and $I$ is the identity matrix of order $|K|$. Let $ν_p(x)$ denote the $p$-adic valuation of the integer $x$. Using bilinear forms on $K$, we show that a $BH(K,h)$ exists whenever (i) $ν_p(h) \geq \lceil ν_p(\exp(K))/2 \rceil$ for every prime divisor $p$ of $|K|$ and (ii) $ν_2(h) \…
Turyn prove that if a circulant Hadamard matrix of order $n$ exists then $n$ must be of the form $n=4m^{2}$ for some odd integer $m$. In this paper we use the structure constant of Schur ring of $\Z_{2}^{4m^{2}}$ to prove that there is no circulant Hadamard matrix in $\Z_{2}^{4m^{2}}$ except possibly for sequences with Hamming weight $a+b$, with $\frac{m^{2}-m}{2}\leq a\leq\frac{3m^{2}-m}{2}$ and $b=2m^{2}-m-a$ and with $\frac{m^{2}+m}{2}\leq 2m^{2}-a\leq\frac{3m^{2}+m}{2}$ and $b=m+a$.
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 …
In this paper we prove that if there is a regular Paley type partial difference set in an Abelian group $G$ of order $v$, where $v=p_1^{2k_1}p_2^{2k_2}\cdots p_n^{2k_n}$, $n\ge 2$, $p_1$, $p_2$, $\cdots$, $p_n$ are distinct odd prime numbers, then for any $1 \le i \le n$, $p_i$ is congruent to 3 modulo 4 whenever $k_i$ is odd. These new necessary conditions further limit the specific order of an Abelian group $G$ in which there can exist a Paley type partial difference set. Our result is similar to a result on Abelian Hadamard (Menon) difference sets proved by Ray-Chaudhuri and Xiang in 1997.
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。
Thin divisible designs graphs: an interplay between fixed-point free involutions of $(v,k,λ)$-graphs and symmetric weighing matrices
本文研究了薄可除设计图中加权矩阵、(v,k,λ)-图及符号图之间的相互作用，提出了两种新的递归构造方法用于正则对称Hadamard矩阵和强正则图，并发现了symplictic图中固定点自由置换的性质。
An $n$ by $n$ skew-symmetric type $(-1,1)$-matrix $K=[k_{i,j}]$ has $1$'s on the main diagonal and $\pm 1$'s elsewhere with $k_{i,j}=-k_{j,i}$. The largest possible determinant of such a matrix $K$ is an interesting problem. The literature is extensive for $n\equiv 0 \mod 4$ (skew-Hadamard matrices), but for $n\equiv 2\mod 4$ there are few results known for this question. In this paper we approach this problem constructing cocyclic matrices over the dihedral group of $2t$ elements, for $t$ odd, which are equivalent to $(-1,1)$-matrices of skew type. Some explicit calculations have been done up…
Determinants of Seidel Tournament Matrices
本文研究锦标赛 Seidel 矩阵行列式平方根所构成的集合，证明其包含关系、区间覆盖、期望与方差、上界及间隙存在性，并建立特征多项式性质。
Nonabelian partial difference sets constructed using abelian techniques
本文利用交换群技巧构造了非交换群中首批阶为$q^{2m}$的部分差集、Paley型部分差集及Paley-Hadamard差集，填补了非交换群PDS研究的空白。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。