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On Fourier and Hadamard matrices of prime order

Primarily AI-generated textHuman understanding: some partsmath.CO — Combinatoricsquant-ph — Quantum Physics

Contributed by Teo Banica ↗

Version 1 / Nov 10, 2026 / CC BY 4.0

Abstract

We study how much of the Fourier matrix FpF_p is needed to recognize it among complex Hadamard matrices of prime order pp. For regular or Butson matrices, taking the ppth power of each entry gives either a Hadamard matrix or a matrix of rank one. For every odd prime, the block of FpF_p on the rows and columns indexed by the nonzero squares determines FpF_p up to equivalence in both classes. At level pp, a symmetry with cycle lengths 1,(p−1)/2,(p−1)/21,(p-1)/2,(p-1)/2 on each side also characterizes FpF_p at every odd prime through 109109.

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Written by Chat GPT, with help from Teo Banica.

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Version history

  1. v1 · 42 pagesSubmitted by Teo BanicaInitial depositCurrentNov 08, 2026