Hadi Kharaghani, Sho Suda, Behruz Tayfeh-Rezaie
The notion of disjoint weighing matrices is introduced as a generalization of orthogonal designs. A recursive construction along with a computer search lead to some infinite classes of disjoint weighing matrices, which in turn are shown to form commutative association schemes with 3 or 4 classes.
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We prove a classification theorem for Hankel weighing matrices.
Inspired by the many applications of mutually unbiased Hadamard matrices, we study mutually unbiased weighing matrices. These matrices are studied for small orders and weights in both the real and complex setting. Our results make use of and examine the sharpness of a very important existing upper bound for the number of mutually unbiased weighing matrices.
We provide a classification method of weighing matrices based on a classification of self-orthogonal codes. Using this method, we classify weighing matrices of orders up to 15 and order 17, by revising some known classification. In addition, we give a revised classification of weighing matrices of weight 5. A revised classification of ternary maximal self-orthogonal codes of lengths 18 and 19 is also presented.
We study the structure of unit weighing matrices of order n and weights 2, 3 and 4. We show that the number of inequivalent unit weighing matrices UW(n,4) depends on the number of decompositions of n into sums of non-negative multiples of some specific positive integers. We also show two interesting sporadic cases in order to show the complexities involved for weights larger than 4.
Unbiased weighing matrices of weight $9$
本文研究权重为9的无偏称重矩阵，提出基于相互适配拉丁方的构造方法，并确定n≤16时相互无偏称重矩阵集合的最大规模，发现最小存在阶数为13，首次找到最大类的阶数为16。
Balanced weighing matrices with parameters $$ \left(1+18\cdot\frac{9^{m+1}-1}{8},9^{m+1},4\cdot 9^m\right), $$ for each nonzero integer $m$ is constructed. This is the first infinite class not belonging to those with classical parameters. It is shown that any balanced weighing matrix is equivalent to a five-class association scheme.
A unified approach to the construction of weighing matrices and certain symmetric designs is presented. Assuming the weight $p$ in a weighing matrix $W(n,p)$ is a prime power, it is shown that there is a $$W\left(\frac{p^{m+1}-1}{p-1}(n-1)+1,p^{m+1}\right)$$ for each positive integer $m$. The case of $n=p+1$ reduces to the balanced weighing matrices with classical parameters $$W\left(\frac{p^{m+2}-1}{p-1},p^{m+1}\right).$$ The equivalence with certain classes of association schemes is discussed in details.
Some mutually quasi-unbiased weighing matrices are constructed from binary codes satisfying certain conditions. Motivated by this, in this note, we study binary codes satisfying the conditions. The weight distributions of binary codes satisfying the conditions are determined. We also give a classification of binary codes of lengths $8,16$ and binary maximal codes of length $32$ satisfying the conditions. As an application, sets of $8$ mutually quasi-unbiased weighing matrices for parameters $(16,16,4,64)$ and $4$ mutually quasi-unbiased weighing matrices for parameters $(32,32,4,256)$ are cons…
A weighing matrix $W$ is quasi-balanced if $|W||W|^\top=|W|^\top|W|$ has at most two off-diagonal entries, where $|W|_{ij}=|W_{ij}|$. A quasi-balanced weighing matrix $W$ signs a strongly regular graph if $|W|$ coincides with its adjacency matrix. Among other things, signed strongly regular graphs and their equivalent association schemes are presented.
Orthogonal designs and weighing matrices have many applications in areas such as coding theory, cryptography, wireless networking and communication. In this paper, we first show that if positive integer $k$ cannot be written as the sum of three integer squares, then there does not exist any skew-symmetric weighing matrix of order $4n$ and weight $k$, where $n$ is an odd positive integer. Then we show that for any square $k$, there is an integer $N(k)$ such that for each $n\ge N(k)$, there is a symmetric weighing matrix of order $n$ and weight $k$. Moreover, we improve some of the asymptotic ex…
In this paper we completely classify the circulant weighing matrices of weight 16 and odd order. It turns out that the order must be an odd multiple of either 21 or 31. Up to equivalence, there are two distinct matrices in CW(31,16), one matrix in CW(21,16) and another one in CW(63,16) (not obtainable by Kronecker product from CW(21,16)). The classification uses a multiplier existence theorem.
A circulant weighing matrix $W = (w_{i,j})$ is a square matrix of order $n$ and entries $w_{i,j}$ in $\{0, \pm 1\}$ such that $WW^T=kI_n$. In his thesis, Strassler gave a table of existence results for such matrices with $n \leq 200$ and $k \leq 100$. In the latest version of Strassler's table given by Tan \cite{arXiv:1610.01914} there are 34 open cases remaining. In this paper we give nonexistence proofs for 12 of these cases, report on preliminary searches outside Strassler's table, and characterize the known proper circulant weighing matrices.
Mutually unbiased weighing matrices (MUWM) are closely related to an antipodal spherical code with 4 angles. In the present paper, we clarify the relationship between MUWM and the spherical sets, and give the complete solution about the maximum size of a set of MUWM of weight 4 for any order. Moreover we describe some natural generalization of a set of MUWM from the viewpoint of spherical codes, and determine several maximum sizes of the generalized sets. They include an affirmative answer of the problem of Best, Kharaghani, and Ramp.
Linear codes with complementary duals are linear codes whose intersection with their duals are trivial, shortly named LCD codes. In this paper we outline a construction for LCD codes over finite fields of order $q$ using weighing matrices and their orbit matrices. The LCD codes constructed can be of any length dimension according to the choice of matrices used in their construction. As a special case, LCD codes of length $2n$ and dimension $n$ are constructed which also have the property of being formally self-dual. Alternatively, under a condition depending on $q$ that the codes are not LCD, …
A survey of complex generalized weighing matrices and a construction of quantum error-correcting codes
本文调研复广义称重矩阵的现有研究，扩展其存在条件与构造、整理小参数存在结果，并基于该矩阵构造量子纠错码以证明其应用价值。
On optimal constant weight codes derived from $ω$-circulant balanced generalized weighing matrices
本文利用具有经典参数的$\omega$-循环平衡加权矩阵，构造了长度为$(q^m-1)/(q-1)$、字母表大小为$g+1$的最优常重码，其中$q$为奇素数幂，$g$整除$q-1$。
The notion of unbiased orthogonal designs is introduced as a generalization among unbiased Hadamard matrices, unbiased weighing matrices and quasi-unbiased weighing matrices. We provide upper bounds and several constructions for mutually unbiased orthogonal designs. As an application, mutually quasi-unbiased weighing matrices for various parameters are obtained.
Constructions of $t$-designs from weighing matrices and association schemes
本文提出通过权矩阵和关联方案构造t-设计的方法，展示了利用会议矩阵生成3-设计的可能性，并探讨了满足特定组合限制的矩阵变体，发现部分平衡不完全块设计和正则配对平衡设计的无限族。
Gauss periods taking exactly two values are closely related to two-weight irreducible cyclic codes and strongly regular Cayley graphs. They have been extensively studied in the work of Schmidt and White and others. In this paper, we consider the question of when Gauss periods take exactly three rational values. We obtain numerical necessary conditions for Gauss periods to take exactly three rational values. We show that in certain cases, the necessary conditions obtained are also sufficient. We give numerous examples where the Gauss periods take exactly three vlaues. Furthermore, we discuss co…
Constructing, Classifying and Studying the Space of Small Integer Weighing Matrices
本文研究并分类了小尺寸n×n权为k的整数权矩阵空间，包括到Hadamard等价和自同构群的不等价矩阵列表，并应用于射影空间权矩阵，提供算法和Sagemath实现，解决了一些开放问题。
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 algorithm for constructing and classifying the space of small integer weighing matrices
本文提出一种在个人计算机上高效构造并分类小尺寸整数称重矩阵的算法，改进了整数平方和表示算法，并给出 $IW(7,25)$ 的完整分类，为寻找经典称重矩阵 $W(35,25)$ 提供基础。
We investigate the existence problem of group invariant matrices using algebraic approaches. We extend the usual concept of multipliers to group rings with cyclotomic integers as coefficients. This concept is combined with the field descent method and rational idempotents to develop new non-existence results.
Complex generalised weighing matrices in centraliser algebras of monomial representations
研究复广义加权矩阵\(CGW(n,w;k)\)，利用有限群单项式表示的中心化子代数，通过对舒尔覆盖线性特征的穷举搜索，对特定条件的此类矩阵分类，恢复相关无限族，构造量子纠错码并确定其最小距离。
Quantum computation in the one-way model requires the preparation of certain resource states known as cluster states. We describe how the construction of continuous-variable cluster states for optical quantum computing relate to the existence of certain families of matrices. The relevant matrices are known as weighing matrices, with a few additional constraints. We prove some results regarding the structure of these matrices, and their associated graphs.
Cyclic relative difference sets and circulant weighing matrices
本文扩展了对循环相对差集的搜索，并将其应用于循环权矩阵的存在性研究。
Construction of Generalized Weighing-Hadamard Matrices over Finite Fields
研究有限域上广义加权哈达玛矩阵，探讨其存在性、性质与构造，证明可逆矩阵子集构成群，引入等价概念及证明商群是阿贝尔群，并讨论其在编码理论中的应用。
Classification of Symmetric Hadamard Matrices Up to Order 32
本文对阶数≤32的对称阿达马矩阵按阿达马等价分类，修正了阶数28分类的错误，用新算法完成分类与枚举，还扩展算法到称重矩阵并建立相关理论结果。
In this article we investigate how we can employ the structure of combinatorial objects like Hadamard matrices and weighing matrices to device new quantum algorithms. We show how the properties of a weighing matrix can be used to construct a problem for which the quantum query complexity is ignificantly lower than the classical one. It is pointed out that this scheme captures both Bernstein & Vazirani's inner-product protocol, as well as Grover's search algorithm. In the second part of the article we consider Paley's construction of Hadamard matrices, which relies on the properties of quadrati…
Constructions of self-orthogonal and LCD subspace codes
本文研究自正交与LCD子空间码的构造问题，提出从满足特定条件的矩阵集合构造这两类子空间码的一般方法，并基于相互拟无偏称重矩阵、链接设计系统及Deza图等给出具体构造。
Character and Multiplier Obstructions for Circulant Weighing Matrices
该研究结合收缩、特征计算、乘子方法等，证明了8个不同阶数和权重的循环称重矩阵不存在。
A $Cretan(4t+1)$ matrix, of order $4t+1$, is an orthogonal matrix whose elements have moduli $\leq 1$. The only $Cretan(4t+1)$ matrices previously published are for orders 5, 9, 13, 17 and 37. This paper gives infinitely many new $Cretan(4t+1)$ matrices constructed using $regular~Hadamard$ matrices, $SBIBD(4t+1,k,λ)$, weighing matrices, generalized Hadamard matrices and the Kronecker product. We introduce an inequality for the radius and give a construction for a Cretan matrix for every order $n \geq 3$.
The concept of S-permutation matrix is considered. A general formula for counting all disjoint pairs of $n^2 \times n^2$ S-permutation matrices as a function of the positive integer $n$ is formulated and proven in this paper. To do that, the graph theory techniques have been used. It has been shown that to count the number of disjoint pairs of $n^2 \times n^2$ S-permutation matrices, it is sufficient to obtain some numerical characteristics of all $n\times n$ bipartite graphs.
In this work, we propose an optimization approach for constructing various classes of circulant combinatorial designs that can be defined in terms of autocorrelations. The problem is formulated as a so-called feasibility problem having three sets, to which the Douglas-Rachford projection algorithm is applied. The approach is illustrated on three different classes of circulant combinatorial designs: circulant weighing matrices, D-optimal matrices, and Hadamard matrices with two circulant cores. Furthermore, we explicitly construct two new circulant weighing matrices, a $CW(126,64)$ and a $CW(19…
The concept of S-permutation matrix is considered in this paper. It defines when two binary matrices are disjoint. For an arbitrary $n^2 \times n^2$ S-permutation matrix, a lower band of the number of all disjoint with it S-permutation matrices is found. A formula for counting a lower band of the number of all disjoint pairs of $n^2 \times n^2$ S-permutation matrices is formulated and proven. As a consequence, a lower band of the probability of two randomly generated S-permutation matrices to be disjoint is found. In particular, a different proof of a known assertion is obtained in the work. T…
Some numerical characteristics of bipartite graphs in relation to the problem of finding all disjoint pairs of S-permutation matrices in the general $n^2 \times n^2$ case are discussed in this paper. All bipartite graphs of the type $g=<R_g \cup C_g, E_g>$, where $|R_g |=|C_g |=2$ or $|R_g |=|C_g |=3$ are provided. The cardinality of the sets of mutually disjoint S-permutation matrices in both the $4 \times 4$ and $9 \times 9$ cases are calculated.
A Formal Orthogonal Pair is a pair $(A,B)$ of symbolic rectangular matrices such that $AB^T=0$. It can be applied for the construction of Hadamard and Weighing matrices. In this paper we introduce a systematic way for constructing such pairs. Our method involves Representation Theory and Group Cohomology. The orthogonality property is a consequence of non-vanishing maps between certain cohomology groups. This construction has strong connections to the theory of Association Schemes and (weighted) Coherent Configurations. Our techniques are also capable for producing (anti-) amicable pairs. A ha…
A universal adjacency matrix of a graph $G$ with adjacency matrix $A$ is any matrix of the form $U = αA + βI + γJ + δD$ with $α\neq 0$, where $I$ is the identity matrix, $J$ is the all-ones matrix and $D$ is the diagonal matrix with the vertex degrees. In the case that $G$ is the disjoint union of regular graphs, we present an expression for the characteristic polynomials of the various universal adjacency matrices in terms of the characteristic polynomials of the adjacency matrices of the components. As a consequence we obtain a formula for the characteristic polynomial of the Seidel matrix o…
This work examines the concept of S-permutation matrices, namely $n^2 \times n^2$ permutation matrices containing a single 1 in each canonical $n \times n$ subsquare (block). The article suggests a formula for counting mutually disjoint pairs of $n^2 \times n^2$ S-permutation matrices in the general case by restricting this task to the problem of finding some numerical characteristics of the elements of specially defined for this purpose factor-set of the set of $n \times n$ binary matrices. The paper describe an algorithm that solves the main problem. To do that, every $n\times n$ binary matr…
In [Journal of Statistical Planning and Inference (141) (2011) 3697-3704], Roberto Fontana offers an algorithm for obtaining Sudoku matrices. Introduced by Geir Dahl concept disjoint pairs of S-permutation matrices [Linear Algebra and its Applications (430) (2009) 2457-2463] is used in this algorithm. Analyzing the works of G. Dahl and R. Fontana, the question of finding a general formula for counting disjoint pairs of $n^2 \times n^2$ S-permutation matrices as a function of the integer $n$ naturally arises. This is an interesting combinatorial problem that deserves its consideration. The pres…
This work examines the problem to describe an efficient algorithm for obtaining $n^2 \times n^2$ Sudoku matrices. For this purpose, we define the concepts of $n\times n$ $Π_n$-matrix and disjoint $Π_n$-matrices. The article, using the set-theoretical approach, describes an algorithm for obtaining $n^2$-tuples of $n\times n$ mutually disjoint $Π_n$ matrices. We show that in input $n^2$ mutually disjoint $Π_n$ matrices, it is not difficult to receive a Sudoku matrix.
Disjoint pairs in set systems and combinatorics of low rank matrices
本文完全解决了Daykin-Erdős关于集族中不相交集合对数量的问题，证明其最优依赖关系，并以强定量形式证明了Singer-Sudan猜想，同时给出对数秩猜想已知界的最优推广。
We study the effects of imposing linear relations within modules of matrices on average sizes of kernels. The relations that we consider can be described combinatorially in terms of partial colourings of grids. The cells of these grids correspond to positions in matrices and each defining relation involves all cells of a given colour. We prove that imposing such relations arising from "admissible" partial colourings has no effect on average sizes of kernels over finite quotients of discrete valuation rings. This vastly generalises the known fact that average sizes of kernels of general square …
Row graphs of Toeplitz matrices
本文研究Toeplitz矩阵的行图，从无三角形行图的Toeplitz矩阵入手推导其最大行和与图结构特征，进一步刻画了几类特殊(0,1)-Toeplitz矩阵的行图性质，明确了行图为单圈的(0,1)-Toeplitz矩阵的特征。
Interchange graphs of (0,1)-matrices are maximally Hamiltonian
研究(0,1)矩阵交换图G(R,S)的哈密顿性，通过结构归纳等方法证明其是最大哈密顿图，二分图时哈密顿可迹，非二分图时哈密顿连通，完整论证经机器验证，为布鲁阿尔迪猜想提供更强结论。
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