This is stuck in directly between " This suggests using a . . . " and " . . . resolvent cubic whose roots may be variously described as a discrete Fourier transform or a Hadamard matrix transform . . ."
32.
To get the punctured Hadamard code above with n = 2 ^ { k-1 }, the chosen Hadamard matrix " H " has to be of Sylvester type, which gives rise to a message length of \ log _ 2 ( 2n ) = k.
33.
A generalization of Sylvester's construction proves that if H _ n and H _ m are Hadamard matrices of orders " n " and " m " respectively, then H _ n \ otimes H _ m is a Hadamard matrix of order " nm ".
34.
Note that any complex Hadamard matrix H can be made into a unitary matrix by multiplying it by \ frac { 1 } { \ sqrt { N } }; conversely, any unitary matrix whose entries all have modulus \ frac { 1 } { \ sqrt { N } } becomes a complex Hadamard upon multiplication by \ sqrt { N }.
35.
In 1933, Raymond Paley discovered the Paley construction, which produces a Hadamard matrix of order " q " + 1 when " q " is any congruent to 3 modulo 4 and that produces a Hadamard matrix of order 2 ( " q " + 1 ) when " q " is a prime power that is congruent to 1 modulo 4.
36.
In 1933, Raymond Paley discovered the Paley construction, which produces a Hadamard matrix of order " q " + 1 when " q " is any congruent to 3 modulo 4 and that produces a Hadamard matrix of order 2 ( " q " + 1 ) when " q " is a prime power that is congruent to 1 modulo 4.
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