Hankel Weighing Matrices

Balanced Weighing Matrices

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.

On binary codes related to mutually quasi-unbiased weighing matrices

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…

Quasi-balanced weighing matrices, signed strongly regular graphs and association schemes

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.

Some non-existence and asymptotic existence results for weighing matrices

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…

The Classification of Circulant Weighing Matrices of Weight 16 and Odd Order

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.

New Nonexistence Results on Circulant Weighing Matrices

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.

Weighing matrices and spherical codes

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.

LCD codes from weighing matrices

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, …

Riordan arrays and the LDU decomposition of symmetric Toeplitz plus Hankel matrices

We examine a result of Basor and Ehrhardt concerning Hankel and Toeplitz plus Hankel matrices, within the context of the Riordan group of lower-triangular matrices. This allows us to determine the LDU decomposition of certain symmetric Toeplitz plus Hankel matrices. We also determine the generating functions and Hankel transforms of associated sequences.

复广义称重矩阵综述及量子纠错码的一种构造

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。

由$\omega$-循环平衡广义加权矩阵导出的最优常重码

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$q-1$。

Unbiased orthogonal designs

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.

从权矩阵和关联方案构造t-设计

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。

Three-valued Gauss periods, circulant weighing matrices and association schemes

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…

构造、分类和研究小整数权矩阵的空间

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。

Symmetric, Hankel-symmetric, and Centrosymmetric Doubly Stochastic Matrices

We investigate convex polytopes of doubly stochastic matrices having special structures: symmetric, Hankel symmetric, centrosymmetric, and both symmetric and Hankel symmetric. We determine dimensions of these polytopes and classify their extreme points. We also determine a basis of the real vector spaces generated by permutation matrices with these special structures.

薄可除设计图：固定点自由置换与(v,k,λ)-图及对称加权矩阵的相互作用

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。

一种用于构造和分类小整数称重矩阵空间的算法

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。

Group Invariant Weighing Matrices

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.

单项式表示的中心化子代数中的复广义加权矩阵

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。

Weighing matrices and optical quantum computing

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.

对称 Toeplitz 矩阵与 Hankel 矩阵的结构性质

Structural properties of a symmetric Toeplitz and Hankel matrices

该研究引入加权 Toeplitz 图与加权 Hankel 图，通过分析图的组成部分，证明两类矩阵的 Frobenius 标准形均可分解为不可约矩阵的直和。

循环相对差集与循环权矩阵

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。

Relatively Prime Polynomials and Nonsingular Hankel Matrices over Finite Fields

The probability for two monic polynomials of a positive degree n with coefficients in the finite field F_q to be relatively prime turns out to be identical with the probability for an n x n Hankel matrix over F_q to be nonsingular. Motivated by this, we give an explicit map from pairs of coprime polynomials to nonsingular Hankel matrices that explains this connection. A basic tool used here is the classical notion of Bezoutian of two polynomials. Moreover, we give simpler and direct proofs of the general formulae for the number of m-tuples of relatively prime polynomials over F_q of given degr…

Permanental ideals of Hankel matrices

We provide the Grobner basis and the primary decomposition of the ideals generated by 2 by 2 permanents of Hankel matrices.

有限域上广义加权哈达玛矩阵的构造

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。

关于与类 Catalan 数相关序列的 Hankel 行列式的若干结果与猜想

Some results and conjectures about Hankel determinants of sequences which are related to Catalan-like numbers

本文研究类 Catalan 数对应容许矩阵各列序列的 Hankel 行列式，归纳计算机实验揭示的性质，并通过前置零行探讨后向移位情形。

阶数不超过32的对称阿达马矩阵的分类

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。

Hankel Matrices for the Period-Doubling Sequence

We give an explicit evaluation, in terms of products of Jacobsthal numbers, of the Hankel determinants of order a power of two for the period-doubling sequence. We also explicitly give the eigenvalues and eigenvectors of the corresponding Hankel matrices. Similar considerations give the Hankel determinants for other orders.

Quantum Algorithms for Weighing Matrices and Quadratic Residues

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…

某些生产矩阵的输出矩阵的行生成多项式的系数式Hankel全正性

Coefficientwise Hankel-total positivity of the row-generating polynomials for the output matrices of certain production matrices

本文利用生产矩阵理论，给出了广义m-Jacobi-Rogers三角形等行生成多项式的系数式Hankel全正性判别准则，并应用于多种组合多项式，解决了Sokal提出的猜想。

Riordan 阵列与 OEIS A005773 的移位 Hankel 行列式

Riordan Arrays and Shifted Hankel Determinants of OEIS A005773

本文通过边界分解统一两个 Riordan 阵列，得到 Motzkin 端点分布的周期三变换及按高度模 3 的路径公式，并导出第三、第四 Hankel 行列式的闭式解。

The determinantal ideals of extended Hankel matrices

In this paper, we use the tools of Gröbner bases and combinatorial secant varieties to study the determinantal ideals $I_t$ of the extended Hankel matrices. Denote by $c$-chain a sequence $a_1,\...,a_k$ with $a_i+c<a_{i+1}$ for all $i=1,\...,k-1$. Using the results of $c$-chain, we solve the membership problem for the symbolic powers $I_t^{(s)}$ and we compute the primary decomposition of the product $I_{t_1}\... I_{t_k}$ of the determinantal ideals. Passing through the initial ideals and algebras we prove that the product $I_{t_1}\... I_{t_k}$ has a linear resolution and the multi-homogeneous…

Hankel determinants of harmonic numbers and related topics

We give an overview of known results about Hilbert matrices from the point of view of orthogonal polynomials and compute Hankel determinants of harmonic numbers and related topics.

Orthogonal polynomials and Smith normal form

Smith normal form evaluations found by Bessenrodt and Stanley for some Hankel matrices of q-Catalan numbers are proven in two ways. One argument generalizes the Bessenrodt-Stanley results for the Smith normal form of a certain multivariate matrix that refines one studied by Berlekamp, Carlitz, Roselle, and Scoville. The second argument, which uses orthogonal polynomials, generalizes to a number of other Hankel matrices, Toeplitz matrices, and Gram matrices. It gives new results for q-Catalan numbers, q-Motzkin numbers, q-Schröder numbers, q-Stirling numbers, q-matching numbers, q-factorials, q…

循环称重矩阵的新不存在性结果

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。

通过Schur多项式高效验证Hankel矩阵的k-符号一致性

Efficient $k$-Sign Consistency Verification of Hankel Matrices via Schur Polynomials

通过Schur多项式高效验证Hankel矩阵的k-符号一致性，利用非负整数线性组合和Littlewood--Richardson系数证明其非负性。

Some determinants of path generating functions, II

We evaluate Hankel determinants of matrices in which the entries are generating functions for paths consisting of up-steps, down-steps and level steps with a fixed starting point but variable end point. By specialisation, these determinant evaluations have numerous corollaries. In particular, one consequence is that the Hankel determinant of Motzkin prefix numbers equals 1, regardless of the size of the Hankel matrix.

A feasibility approach for constructing combinatorial designs of circulant type

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…

Spectral measure of large random Hankel, Markov and Toeplitz matrices

We study the limiting spectral measure of large symmetric random matrices of linear algebraic structure. For Hankel and Toeplitz matrices generated by i.i.d. random variables $\{X_k\}$ of unit variance, and for symmetric Markov matrices generated by i.i.d. random variables $\{X_{ij}\}_{j>i}$ of zero mean and unit variance, scaling the eigenvalues by $\sqrt{n}$ we prove the almost sure, weak convergence of the spectral measures to universal, nonrandom, symmetric distributions $γ_H$, $γ_M$ and $γ_T$ of unbounded support. The moments of $γ_H$ and $γ_T$ are the sum of volumes of solids related to …

Finding linearly generated subsequences

We develop a new algorithm to compute determinants of all possible Hankel matrices made up from a given finite length sequence over a finite field. Our algorithm fits within the dynamic programming paradigm by exploiting new recursive relations on the determinants of Hankel matrices together with new observations concerning the distribution of zero determinants among the possible matrix sizes allowed by the length of the original sequence. The algorithm can be used to isolate \emph{very} efficiently linear shift feedback registers hidden in strings with random prefix and random postfix for ins…

Square ice, alternating sign matrices and classical orthogonal polynomials

The six-vertex model with Domain Wall Boundary Conditions, or square ice, is considered for particular values of its parameters, corresponding to 1-, 2-, and 3-enumerations of Alternating Sign Matrices (ASMs). Using Hankel determinant representations for the partition function and the boundary correlator of homogeneous square ice, it is shown how the ordinary and refined enumerations can be derived in a very simple and straightforward way. The derivation is based on the standard relationship between Hankel determinants and orthogonal polynomials. For the particular sets of parameters correspon…

有限域上Hankel矩阵的秩研究

On the rank of Hankel matrices over finite fields

本研究细化了有限域上秩不超过r的Hankel矩阵计数的经典结果，证明固定前k个元素时满足秩条件的剩余元素选择数为q^(2r−k)，并推广了有限域上Jacobi-Trudi行列式求值的相关结论。