Editing Hadamard matrix (section)

===Alternative construction===
If we map the elements of the Hadamard matrix using the [[group homomorphism]] <math> (\{1, -1\}, \times) \rightarrow (\{0, 1\}), +) </math>, where <math>(\{0, 1\}), +) </math> is the additive group of the [[finite field|field]] <math>\mathrm{GF}(2)</math> with two elements, we can describe an alternative construction of Sylvester's Hadamard matrix. First consider the matrix <math> F_n </math>, the <math> n\times 2^n </math> matrix whose columns consist of all ''n''-bit numbers arranged in ascending counting order. We may define <math> F_n </math> recursively by

:<math>\begin{align}
  F_1 &= \begin{bmatrix}0 & 1\end{bmatrix} \\
  F_n &= \begin{bmatrix}
    0_{1\times 2^{n-1}} & 1_{1\times 2^{n-1}} \\
    F_{n-1}             & F_{n-1}
  \end{bmatrix}.
\end{align}</math>

It can be shown by [[mathematical induction|induction]] that the image of the Hadamard matrix under the above homomorphism is given by

: <math>H_{2^n} \mapsto F_n^\textsf{T} F_n,</math>

where the matrix arithmetic is done over <math>\mathrm{GF}(2)</math>.

This construction demonstrates that the rows of the Hadamard matrix <math> H_{2^n} </math> can be viewed as a length <math> 2^n </math> linear [[error-correcting code]] of [[linear code#Popular notation|rank]] ''n'', and [[linear code#Properties|minimum distance]] <math> 2^{n-1} </math> with [[linear code#Popular notation|generating matrix]] <math> F_n. </math>

This code is also referred to as a [[Walsh code]]. The [[Hadamard code]], by contrast, is constructed from the Hadamard matrix <math> H_{2^n} </math> by a slightly different procedure.