Hadamard Conjecture Proof

EasyChair Preprint 8250

Abstract

The most important open question in the theory of Hadamard matrices is that of existence (the Hadamard conjecture). The generalization of Sylvester’s construction proves that if H_n and H_m are Hadamard matrices of orders n and m, respectively, then H_n x H_m given by the Kronecker product is a Hadamard matrix of order nm. This result is used to produce Hadamard matrices of higher order once those of smaller orders are known.

Here we go in the opposite direction: we show how to construct a Hadamard matrix of order 4n from a Hadamard matrix of order 4nm. The outcome gives the link to Number Theory with a way to prove the Hadamard conjecture, using Paley’s work.

Keyphrases: Hadamard codes, Hadamard conjecture, Hadamard matrix, Kronecker product, Paley’s work, coding theory, discrete Fourier analysis, integer lattices

@booklet{EasyChair:8250,
  author    = {Valerii Sopin},
  title     = {Hadamard Conjecture Proof},
  howpublished = {EasyChair Preprint 8250},
  year      = {EasyChair, 2022}}