Periodicity of sub columns in Hadamard matrix

Question

Let's consider the Hadamard transform $H_n$ where $H_{ij} = (-1)^{i.j}$. I want to count the number of repeated sub-columns of length $l$ in this matrix. Does it exist any formula or combinatorial tools to tackle this kind of problem? For more clarity let me give a simple example: taking $l=2$, we directly see that the only two possible sub-columns of length 2 are: $c_1=(1,1)^T$ and $c_2=(1,-1)^T$ which both appear $n/2$ times. I suspect that if the length $l > n/2$ then we won't have any repeated such sub-column right?

Answer 1

I may have found the answer by recursion. In order to look at the number of distinct sub-columns of length $l$ in $H_n$, I first build the smallest Hadamard matrix that contains the rows of length $l$, that is, a matrix $H$ of dimension $\lfloor \log_2l\rfloor$ such that this matrix trivially contains my $2^{\lfloor \log_2l\rfloor}$ distinct sub-columns appearing only one time. Next, since $H_n = H_1 \otimes\dots \otimes H_1 \otimes H_{\lfloor \log_2l\rfloor}$ and knowing that each that we apply the tensor product $H_1$ with our matrix it doubles the number of occurrence of our sub-columns, they will appear $2^{n-\lfloor \log_2l\rfloor}$ in $H_n$.