arXiv subjects

Assaf Goldberger

Publications and source records attributed to Assaf Goldberger.

10 recordsLinked to original sources

arXiv2026

Constructing, Classifying and Studying the Space of Small Integer Weighing Matrices

Integer weighing matrices (IW-matrices for short) are integer valued orthogonal square matrices. One usecase of these is to create classical weighing matrices with various block structures. In this paper we study and classify the space $IW(n,k)$ of the integer weighing matrices of small size $n\times n$ and weight $k$. Our classification includes a full list of all inequivalent matrices up to Hadamard equivalence and automorphism groups. We then continue to a secondary classification of the symmetric and antisymmetric IW up to symmetric Hadamard equivalence. We apply this to the case of projective space weighing matrices. Next we use the classification to count the cardinality of the spaces of all $IW(n,k)$ as well as the symmetric and anti-symmetric subspace. We supply practical algorithms and implement them in \texttt{Sagemath}. Finding an (anti-)symmetric IW matrix in a given Hadamard class can be done for significantly higher orders. In particular we solve some open cases: Symmetric $W(23,16)$, $W(28,25)$ and $W(30,17)$, and an anti-symmetric $W(28,25)$. We conclude by showing a detailed classification of $IW(7,25)$. We have also improved the \texttt{NSOKS} algorithm to find all possible representations of an integer $k$ as a sum of $n$ integer squares.

math.CO↗

arXiv2025

Abelian and Dihedral equiangular tight frames of redundancy $2$

This paper studies group frames ($G$-frames) where the unitary group representation can be projective. When the group is abelian, for most combinations $N, n$, we show that $ETF(N,n)$ can only exist for genuinely projective group representations. In particular, cyclic-group frames for such parameters do not exist. We also give a characterization of all dihedral tight frames and dihedral $ETF(2n,n)$, using which, we conclude that regular dihedral $ETF(2n,n)$ must be genuinely projective. Following that, we give a characterization of regular dihedral $ETF(2n,n)$ in terms of certain structured skew Hadamard matrices. We then show that Paley $ETF(2n,n)$ and its doubling are both of this type. Finally, we classify all regular dihedral $ETF(2n,n)$ for $n\le 22$ up to switching equivalence.

math.CO↗

arXiv2023

Cohomology-Developed Matrices -- constructing families of weighing matrices and automorphism actions

The aim of this work is to construct families of weighing matrices via their automorphism group action. This action is determined from the $0,1,2$-cohomology groups of the underlying abstract group. As a consequence, some old and new families of weighing matrices are constructed. These include the Paley Conference, the Projective-Space, the Grassmannian, and the Flag-Variety weighing matrices. We develop a general theory relying on low dimensional group-cohomology for constructing automorphism group actions, and in turn obtain structured matrices that we call \emph{Cohomology-Developed matrices}. This "Cohomology-Development" generalizes the Cocyclic and Group Developments. The Algebraic structure of modules of Cohomology-Developed matrices is discussed, and an orthogonality result is deduced. We also use this algebraic structure to define the notion of \emph{quasiproducts}, which is a generalization of the Kronecker-product.

math.GR↗

arXiv2023

An algorithm for constructing and classifying the space of small integer weighing matrices

In this paper we describe an algorithm for generating all the possible $PIW(m,n,k)$ - integer $m\times n$ Weighing matrices of weight $k$ up to Hadamard equivalence. Our method is efficient on a personal computer for small size matrices, up to $m\le n=12$, and $k\le 50$. As a by product we also improved the \textit{\textbf{nsoks}} \cite{riel2006nsoks} algorithm to find all possible representations of an integer $k$ as a sum of $n$ integer squares. We have implemented our algorithm in \texttt{Sagemath} and as an example we provide a complete classification for \ $n=m=7$ and $k=25$. Our list of $IW(7,25)$ can serve as a step towards finding the open classical weighing matrix $W(35,25)$.

math.CO↗

arXiv2022

Phase transitions for frame potentials]{Phase transitions for the minimizers of the $p^{th}$ frame potentials in $\mathbb{R}^2$

Given $N$ points $X=\{x_k\}_{k=1}^N$ on the unit circle in $\mathbb{R}^2$ and a number $0\leq p \leq \infty$ we investigate the minimizers of the functional $\sum_{k, \ell =1}^N |\langle x_k, x_\ell\rangle|^p$. While it is known that each of these minimizers is a spanning set for $\mathbb{R}^2$, less is known about their number as a function of $p$ and $N$ especially for relatively small $p$. In this paper we show that there is unique minimum for this functional for all $p\leq \log 3/\log 2$ and all odd $N\geq 3$. In addition, we present some numerical results suggesting the emergence of a phase transition phenomenon for these minimizers. More specifically, for $N\geq 3$ odd, there exists a sequence of number of points $\log 3/\log 2=p_1< p_2< \cdots < p_N\leq 2$ so that a unique (up to some isometries) minimizer exists on each sub-intervals $(p_k, p_{k+1})$. %In addition we conjecture that $\lim_{k\to \infty}p_{2k+1}=2$.

math.CO↗

arXiv2021

Towards a classification of incomplete Gabor POVMs in $\mathbb{C}^d$

Every (full) finite Gabor system generated by a unit-norm vector $g\in \mathbb{C}^d$ is a finite unit-norm tight frame (FUNTF), and can thus be associated with a (Gabor) positive operator valued measure (POVM). Such a POVM is informationally complete if the $d^2$ corresponding rank one matrices form a basis for the space of $d\times d$ matrices. A sufficient condition for this to happen is that the POVM is symmetric, which is equivalent to the fact that the associated Gabor frame is an equiangular tight frame (ETF). The existence of Gabor ETF is an important special case of the Zauner conjecture. It is known that generically all Gabor FUNTFs lead to informationally complete POVMs. In this paper, we initiate a classification of non-complete Gabor POVMs. In the process we establish some seemingly simple facts about the eigenvalues of the Gram matrix of the rank one matrices generated by a finite Gabor frame. We also use these results to construct some sets of $d^2$ unit vectors in $\mathbb{C}^d$ with a relatively smaller number of distinct inner products.

math.FA↗

arXiv2020

Formal Orthogonal Pairs via Monomial Representations and Cohomology

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 handful of examples are given.

math.CO↗

arXiv2018

Energy Minimization in $CP^n$: Some Numerical and Analytical Results

We study the problem of minimizing the energy function $M^p(m,n) := \min \sum_{1\le i 0$ are integers and $p$ is even. This problem has implications on finding nice polyhedra in projective spaces, and on quantum random access codes. We conduct experimental search in the complex case which suggests nice patterns on the minimum values. In some cases($p=2$ and partially $n=2$) we supply analytical proofs and give full descriptions of the minimal configurations. We also show that as $m\to \infty$, nearly equidistributed configurations points nearly give the minimal values we expect from our patterns.

math.MG↗

arXiv2015

On the finite geometry of $W(23,16)$

We study the local geometry of the zero pattern of a weighing matrix $W(23,16)$. The geometry consists of $23$ lines and $23$ points where each line contains $7$ points. The incidence rules are that every two lines intersect in an odd number of points, and the dual statement holds as well. We show that more than $50\%$ of the pairs of lines must intersect at a single point, and construct a regular weighted graph out of this geometry. This might indicate that a weighing matrix $W(23,16)$ does not exist.

math.CO↗

arXiv2015

Diagonals of real symmetric matrices of given spectra as a measure space

The set of diagonals of real symmetric matrices of given non negative spectrum is endowed with a measure which is obtained by the push forward of the Haar measure of the real orthogonal group.\\ We prove that the Radon Nicodym derivation of this measure with respect to the relative Euclidean measure is approximated by the coefficients of a sequence of zonal sphere polynomials corresponding with the given spectrum. There is a striking similarity between the role of the zonal sphere polynomials in the orthogonal case, and that of the Schur function in the Hermitian case.\\ Following this we obtain a combinatorial approximation for the probability of real symmetric matrix of a given spectrum to appear as the sum of two real symmetric matrices, each of a given spectrum. In addition we obtain a real orthogonal analogue to the Zuber Itzykson Harish Chandra integration formula.

math.RT↗