TC08 / 6. Hadamard codes 3.12.07 SX
Hadamard matrices. Paley’s construction of Hadamard matrices Hadamard codes. Decoding Hadamard codes
Hadamard matrices A Hadamard matrix of order is a matrix of type whose coefficients are1 or 1 and such that . Note that this relation is equivalent to say that the rows of have norm √ and that any two distinct rows are orthogonal. Since is invertible and , we see that 2 and hence is also a Hadamard matrix. Thus is a Hadamard matrix if and only if its columns have norm √ and any two distinct columns are orthogonal.
If is a Hadamard matrix of order , then det ⁄ . In particular we see that the Hadamard matrices satisfy the equality in Hadamard’s inequality: