Related papers: Butson Hadamard matrices, bent sequences, and spherical codes

Butson Hadamard matrices, bent sequences, and spherical codes

URL: http://arxiv.org/abs/2311.00354v1
Date: Wed, 1 Nov 2023 08:03:11 GMT
Title: Butson Hadamard matrices, bent sequences, and spherical codes
Authors: Minjia Shi, Danni Lu, Andrés Armario, Ronan Egan, Ferruh Ozbudak, Patrick Solé,
Abstract summary: We explore a notion of bent sequence attached to the data consisting of an Hadamard matrix of order $n$ defined over the complex $qth$ roots of unity. In particular we construct self-dual bent sequences for various $qle 60$ and lengths $nle 21.$ construction methods comprise the resolution of systems by Groebner bases and eigenspace computations.
Score: 15.98720468046758
License: http://creativecommons.org/licenses/by/4.0/
Abstract: We explore a notion of bent sequence attached to the data consisting of an Hadamard matrix of order $n$ defined over the complex $q^{th}$ roots of unity, an eigenvalue of that matrix, and a Galois automorphism from the cyclotomic field of order $q.$ In particular we construct self-dual bent sequences for various $q\le 60$ and lengths $n\le 21.$ Computational construction methods comprise the resolution of polynomial systems by Groebner bases and eigenspace computations. Infinite families can be constructed from regular Hadamard matrices, Bush-type Hadamard matrices, and generalized Boolean bent functions.As an application, we estimate the covering radius of the code attached to that matrix over $\Z_q.$ We derive a lower bound on that quantity for the Chinese Euclidean metric when bent sequences exist. We give the Euclidean distance spectrum, and bound above the covering radius of an attached spherical code, depending on its strength as a spherical design.

Related papers

Understanding Matrix Function Normalizations in Covariance Pooling through the Lens of Riemannian Geometry [63.694184882697435]
Global Covariance Pooling (GCP) has been demonstrated to improve the performance of Deep Neural Networks (DNNs) by exploiting second-order statistics of high-level representations.
arXiv Detail & Related papers (2024-07-15T07:11:44Z)
A note on MDS Property of Circulant Matrices [3.069335774032178]
In $2014$, Gupta and Ray proved that the circulant involutory matrices over the finite field $mathbbF_2m$ can not be maximum distance separable (MDS) This article delves into circulant matrices possessing these characteristics over the finite field $mathbbF_2m$.
arXiv Detail & Related papers (2024-06-22T16:00:00Z)
On MDS Property of g-Circulant Matrices [3.069335774032178]
We first discuss $g$-circulant matrices with involutory and MDS properties. We then delve into $g$-circulant semi-involutory and semi-orthogonal matrices with entries from finite fields.
arXiv Detail & Related papers (2024-06-22T15:18:31Z)
Efficient Unitary T-designs from Random Sums [0.6640968473398456]
Unitary $T$-designs play an important role in quantum information, with diverse applications in quantum algorithms, benchmarking, tomography, and communication. We provide a new construction of $T$-designs via random matrix theory using $tildeO(T2 n2)$ quantum gates.
arXiv Detail & Related papers (2024-02-14T17:32:30Z)
One-sided Matrix Completion from Two Observations Per Row [95.87811229292056]
We propose a natural algorithm that involves imputing the missing values of the matrix $XTX$. We evaluate our algorithm on one-sided recovery of synthetic data and low-coverage genome sequencing.
arXiv Detail & Related papers (2023-06-06T22:35:16Z)
Deep Learning Symmetries and Their Lie Groups, Algebras, and Subalgebras from First Principles [55.41644538483948]
We design a deep-learning algorithm for the discovery and identification of the continuous group of symmetries present in a labeled dataset. We use fully connected neural networks to model the transformations symmetry and the corresponding generators. Our study also opens the door for using a machine learning approach in the mathematical study of Lie groups and their properties.
arXiv Detail & Related papers (2023-01-13T16:25:25Z)
An Equivalence Principle for the Spectrum of Random Inner-Product Kernel Matrices with Polynomial Scalings [21.727073594338297]
This study is motivated by applications in machine learning and statistics. We establish the weak limit of the empirical distribution of these random matrices in a scaling regime. Our results can be characterized as the free additive convolution between a Marchenko-Pastur law and a semicircle law.
arXiv Detail & Related papers (2022-05-12T18:50:21Z)
Leverage Score Sampling for Tensor Product Matrices in Input Sparsity Time [54.65688986250061]
We give an input sparsity time sampling algorithm for approximating the Gram matrix corresponding to the $q$-fold column-wise tensor product of $q$ matrices. Our sampling technique relies on a collection of $q$ partially correlated random projections which can be simultaneously applied to a dataset $X$ in total time.
arXiv Detail & Related papers (2022-02-09T15:26:03Z)
Non-PSD Matrix Sketching with Applications to Regression and Optimization [56.730993511802865]
We present dimensionality reduction methods for non-PSD and square-roots" matrices. We show how these techniques can be used for multiple downstream tasks.
arXiv Detail & Related papers (2021-06-16T04:07:48Z)
Algebraic and geometric structures inside the Birkhoff polytope [0.0]
Birkhoff polytope $mathcalB_d$ consists of all bistochastic matrices of order $d$. We prove that $mathcalL_d$ and $mathcalF_d$ are star-shaped with respect to the flat matrix.
arXiv Detail & Related papers (2021-01-27T09:51:24Z)
Quantum algorithms for spectral sums [50.045011844765185]
We propose new quantum algorithms for estimating spectral sums of positive semi-definite (PSD) matrices. We show how the algorithms and techniques used in this work can be applied to three problems in spectral graph theory.
arXiv Detail & Related papers (2020-11-12T16:29:45Z)

This list is automatically generated from the titles and abstracts of the papers in this site.