$H$ is an $n \times n$ matrix with elements in $ \{ -1,0,1 \}$
$G$ is an $n \times k$ matrix with elements in $GF(2)$ and also upper triangular, invertable
$m$ is an $k \times 1$ vector with elements in $GF(2)$
How can we perceive the output of $HGm$ where $Gm$ multiplication is in $GF(2)$ and $H$ multiplication is a normal real multiplication. Actually I want to combine $HG$ transformation into one $P$ transformation. How can I multiply two matrices while elements in one are in $GF(2)$ and other is in $R$ ? (We can also restrict the entries in $H$ to be one of $-1$ and $1$ but the output can be in $R$).
Motivation: It is a digital communication problem. $Gm$ is output codeword with 1 being mapped to -1 and 0 bit being mapped to +1. This codeword is multiplied to a channel convolution matrix $H$ e.g.
write in MATLAB
H = [1 -1 0 0 0;0 1 -1 0 0; 0 0 1 -1 0; 0 0 0 1 -1; 0 0 0 0 1]'
Now I have the following questions?
I want to know whether the whole transformation P = H(G) can be represented by an equivalent transformation P' in GF(2) ? In other words, does there exist a $M \in [GF(2)]^{n \times n}$ such that P'=G; P=H(MG)
If above is not generally possible, do there exist sufficient conditions for $H$ such that above is even possible?
Which branch of mathematics deals with the interaction of real matrices with matrices on finite fields?