Hadamard's maximal determinant problem [SEP] named after
Hadamard's maximal determinant problem
Hadamard's maximal determinant problem, named after Jacques Hadamard, asks for the largest determinant of a matrix with elements equal to 1 or −1. The analogous question for matrices with elements equal to 0 or 1 is equivalent since, as will be shown below, the maximal determinant of a {1,−1} matrix of size "n" is 2 times the maximal determinant of a {0,1} matrix of size "n"−1. The problem was posed by Hadamard in the 1893 paper in which he presented
well as rows. The "n"-dimensional parallelotope spanned by the rows of an "n"×"n" Hadamard matrix has the maximum possible "n"-dimensional volume among parallelotopes spanned by vectors whose entries are bounded in absolute value by 1. Equivalently, a Hadamard matrix has maximal determinant among matrices with entries of absolute value less than or equal to 1 and so is an extremal solution of Hadamard's maximal determinant problem.
Certain Hadamard matrices can almost directly be used as an error-correcting code using a Hadamard code (generalized in Reed–Muller