Hadamard matrix
Any two rows of an "n"×"n" Hadamard matrix are orthogonal. For a {1, −1} matrix, it means any two rows differ in exactly half of the entries, which is impossible when "n" is an odd number. When "n" ≡ 2 (mod 4), two rows that are both orthogonal to a third row cannot be orthogonal to each other. Together, these statements imply that an "n"×"n" Hadamard matrix can exist only if "n" = 1, 2, or a multiple of 4. Hadamard matrices have been well studied, but it is not known whether an "n"×"n" Hadamard matrix exists for every "n" that is a positive multiple of 4. The smallest "n" for which an "n"×"n" Hadamard matrix is not known to exist is 668.