===Construction using general Hadamard matrices===
Generalized Hadamard codes are obtained from an ''n''-by-''n'' [[Hadamard matrix]] ''H''. In particular, the 2''n'' codewords of the code are the rows of ''H'' and the rows of −''H''. To obtain a code over the alphabet {0,1}, the mapping −1 ↦ 1, 1 ↦ 0, or, equivalently, ''x'' ↦ (1 − ''x'')/2, is applied to the matrix elements. That the minimum distance of the code is ''n''/2 follows from the defining property of Hadamard matrices, namely that their rows are mutually orthogonal. This implies that two distinct rows of a Hadamard matrix differ in exactly ''n''/2 positions, and, since negation of a row does not affect orthogonality, that any row of ''H'' differs from any row of −''H'' in ''n''/2 positions as well, except when the rows correspond, in which case they differ in ''n'' positions.
To get the augmented Hadamard code above with <math>n=2^{k-1}</math>, the chosen Hadamard matrix ''H'' has to be of Sylvester type, which gives rise to a message length of <math>\log_2(2n)=k</math>.
