Hadamard's conjecture

Problem's Description

The Hadamard maximal determinant problem is a fascinating combinatorial optimization problem with many concrete applications to signal processing, coding theory, experimental design theory, and cryptography . Let $M$ be an $n\times n$ matrix, all of whose entries are at most 1 in modulus. Note that $det(M)$ is equal to the volume of the $n$-dimensional parallelepiped spanned by the rows of $M$. By assumption, each row has Euclidean length at most $n^{1/2}$. Then $det(M) \leq n^{n/2}$, and equality holds if and only if every entry of $M$ is $\pm 1$ and the rows of $M$ are orthogonal. Hadamard established this upper bound which is only attainable if the order n equals $1, 2,$ or a multiple of $4.$ A matrix attaining the bound is called a Hadamard matrix. The classical and very old Hadamard conjecture asserts that a Hadamard matrix exists of every order divisible by $4.$ The smallest multiple of $4$ for which no such matrix is currently known is $668$, the value $428$ having been settled only in $2005$. Due to its extensive real life applications, it is still a very active line of research.

1. ArticleIs an originR\'esolution d\'une question relative aux d\'eterminants

   • Jacques Hadamard

2. Article A Hadamard matrix of order 428

   • H. Kharaghani,
   • B. Tayfeh-Rezaie

   Journal of Combinatorial Designs 13, 435-440, 2005fulltext