Hadamard Matrix - Alternative Construction

Alternative Construction

If we map the elements of the Hadamard matrix using the group homomorphism, we can describe an alternative construction of Sylvester's Hadamard matrix. First consider the matrix, the matrix whose columns consist of all n-bit numbers arranged in ascending counting order. We may define recursively by

$F_1=\begin{bmatrix} 0 & 1\end{bmatrix}$

$F_n=\begin{bmatrix} 0_{1\times 2^{n-1}} & 1_{1\times 2^{n-1}} \\ F_{n-1} & F_{n-1} \end{bmatrix}.$

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

$H_{2^n}=F_n^{\rm T}F_n.$

This construction demonstrates that the rows of the Hadamard matrix can be viewed as a length linear error-correcting code of rank n, and minimum distance with generating matrix

This code is also referred to as a Walsh code. The Hadamard code, by contrast, is constructed from the Hadamard matrix by a slightly different procedure.

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