Constructing Hadamard matrices

This article discusses Hadamard matrices, which are orthogonal matrices with entries of ±1. It covers Sylvester's construction, Kronecker products, and Paley's method for generating Hadamard matrices of certain orders. The focus is on the set of possible orders and the unresolved Hadamard conjecture, which states that Hadamard matrices exist for all orders divisible by 4. The article highlights a recent breakthrough by Levent Alpöge and collaborators, who constructed a Hadamard matrix of order 668, thereby filling all gaps below 2000. This is significant for both theoretical mathematics and practical applications in coding theory, signal processing, and experimental design.