I'm wondering if there is a general approach to deriving all the possible sets of basis vectors that could fill the entire space of a binary $n\times m$ matrix, where $n \lt m$. That is, is there a general way to determine sets of vectors which could be combined via the binary operators AND, OR, and NOT to realize the full range of binary numbers of length $m$?
For example, say I have some system of $n$ variables in $m$ dimensions, where $n \lt m$, to form a matrix $M$ which represents those variables and all possible permutations of their binary values along each dimension. Without having specific values for any instance of this system, is there an algorithm for deriving sets of possible basis vectors? My apologies in advance if there is some other terminology for this that I am ignorant of or butchering.
Using hand-derived example: Say I have $n=2$ and $m=3$. I want to derive a basis such that any logical combination of a base vector will span all numbers on [$2^0$,$2^3$]. I choose the following basis:
$A= \begin{bmatrix}1&1&0\\0&1&1\end{bmatrix}$
using this, I can achieve all possible values in base 2 of length $m$:
$ \begin{align} 000 &\longrightarrow \lnot 110 \land \lnot 011 \\ 001 &\longrightarrow \lnot 110 \land 011 \\ 010 &\longrightarrow 110 \land 011 \\ 011 &\longrightarrow 011 \\ 100 &\longrightarrow 110 \land \lnot 011\\ 101 &\longrightarrow \lnot 110 \lor \lnot 011\\ 110 &\longrightarrow 110 \\ 111 &\longrightarrow 110 \lor 011 \\ \end{align} $
Thus using this set of 'basis' vectors, I can describe any row variable as a combination of those basis vectors by binary operators.