How to construct Hadamard Matrices of size $3 \cdot 2^n $?

Question

I want to construct Hadamard matrices of size $3 \cdot 2^n$ for values of $n $ like $5,6,10$ etc. So far, I have seen Sylvester's construction that can only construct Hadamard matrices of size $2^n$. Paley's construction can give Hadamard matrix for $q+1$ where $q$ is a prime power. Is there a way to construct Hadamard matrix of size $3 \cdot 2^n$?

Answer 1

1. Construct a $12\times12$ Hadamard matrix (Paley IIRC).
2. Apply the doubling construction ever after.

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The number of rows of a Hadamard matrix is a multiple of four so you need $n\ge2$ anyway, and the above recipe covers all of them.