$H$ is an $n \times n$ matrix with elements in $\{ 0,1 -1,1 \}$

$G$ 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 0. +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 want to impose a restriction on the complete transformation $P=HG$ and want to know if it is upper triangular or not ?

$H$ is an $n \times n$ matrix with elements in $\{ 0,1 \}$

$G$ is an $n \times n$ k$matrix with elements in$GF(2)$and also upper triangular, invertable$m$is an$n 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$0$-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 0. 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 want to impose a restriction on the complete transformation $P=HG$ and want to know if it is upper triangular or not ?

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# Multiplication of matrices in GF(2) and R

$H$ is an $n \times n$ matrix with elements in $\{ 0,1 \}$

$G$ is an $n \times n$ matrix with elements in $GF(2)$

$m$ is an $n \times 1$ 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 $0$ and $1$ but the output can be in $R$).