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Number of solutions to linear system of equations over GF(2)
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Problem
Linear systems of equations over the reals have either 0, 1 or infinitely many solutions. However, when applied to finite fields (specifically GF(2)), infinitely many is not an option.
Is there a fast general method to calculate the number of distinct solutions to a linear system of equations over GF(2)?
You can assume that Gaussian elimination has already been performed, so an example augmented matrix would be:
$$ \left[
\begin{array}{ccccccccc|c}1&0&0&0&1&1&1&0&0&1\\
0&1&0&0&1&0&1&0&1&1\\
0&0&1&0&0&1&1&1&0&0\\
0&0&0&1&1&1&1&1&1&1\\
0&0&0&0&0&0&0&0&0&0\\
0&0&0&0&0&0&0&0&0&0\\
0&0&0&0&0&0&0&0&0&0\\
0&0&0&0&0&0&0&0&0&0\\
0&0&0&0&0&0&0&0&0&0\\
\end{array}
\right] $$
Is there a fast general method to calculate the number of distinct solutions to a linear system of equations over GF(2)?
You can assume that Gaussian elimination has already been performed, so an example augmented matrix would be:
$$ \left[
\begin{array}{ccccccccc|c}1&0&0&0&1&1&1&0&0&1\\
0&1&0&0&1&0&1&0&1&1\\
0&0&1&0&0&1&1&1&0&0\\
0&0&0&1&1&1&1&1&1&1\\
0&0&0&0&0&0&0&0&0&0\\
0&0&0&0&0&0&0&0&0&0\\
0&0&0&0&0&0&0&0&0&0\\
0&0&0&0&0&0&0&0&0&0\\
0&0&0&0&0&0&0&0&0&0\\
\end{array}
\right] $$
Solution
You can take any subset of the free variables, and then correct any resulting sum using the fixed variables.
Thus, after Gaussian elimination the total number of solutions is $2^f$ where $f$ is the number of free variables.
Thus, after Gaussian elimination the total number of solutions is $2^f$ where $f$ is the number of free variables.
Context
StackExchange Computer Science Q#50478, answer score: 2
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