I have matrix Multiplication of $H.Z.X = Y$ as mentioned below :
$\left[\begin{array}{ccc} h_{11} & \dots & h_{N 1} \\ \vdots & \ddots & \vdots \\ h_{N 1} & \cdots & h_{N N} \end{array}\right]\left[\begin{array}{ccc} Z_{11} & \cdots & Z_{N 1} \\ \vdots & \ddots & \vdots \\ Z_{N 1} & \cdots & Z_{N N} \end{array}\right]\left[\begin{array}{cc} X_{1} \\ \cdot \\ x_{N} \end{array}\right]=\left[\begin{array}{c} Y_{1} \\ \cdot \\ y_{N} \end{array}\right]$
The size of matrix $H$ and $Z$ is $(256,256)$ and size of vectors $X$ and $Y$ is $(256,1)$. What I know about matrix $Z$ is permutation matrix where the $32$ first rows are zeros except columns $1,9,17,...(265-8)$. Similarly, next $32$ rows are zeros except columns $2,10,18,...(256-7)$. The next $32$ rows are zeros except columns $3,11,19,...(256-6)$ till the last $32$ rows which are zeros except $8,16,24,...256$.
My question, is it possible to simplify the matrix multiplication by specifying the required part of matrix $H$ to be multiplied with non-zeros values of $Z$? I mean for example get the part of $H_{32,32}$ and multiply it with $Z_{32,32}$ to then multiply it with first $32$ elements of $X_{1:32}$ to be equaled the first $32$ first element of $Y_{1:32}$ and so on.
I mean to avoid the large multiplications of elements with zeros values of matrix $Z$ at the same time to have less complexity in multiplications.