Butson Hadamard matrices, bent sequences, and spherical codes

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 se...

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Bibliographic Details
Published in:arXiv.org 2023-11
Main Authors: Shi, Minjia, Lu, Danni, Armario, Andrés, Egan, Ronan, Ozbudak, Ferruh, Solé, Patrick
Format: Article
Language:English
Subjects:
Automorphisms
Eigenvalues
Euclidean geometry
Lower bounds
Polynomials
Sequences
Online Access:Get full text
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Summary: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.
ISSN:2331-8422