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.

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

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

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Created at: 2019-05-31 14:35:44Z

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