Download PDFOpen PDF in browserHadamard Conjecture ProofEasyChair Preprint no. 82503 pages•Date: June 12, 2022AbstractThe 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: coding theory, discrete Fourier analysis, Hadamard codes, Hadamard conjecture, Hadamard matrix, integer lattices, Kronecker product, Paley’s work
