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Encode: msg="cut", corresponding vector
$m = \left[
\begin{array}{c}
2\\
20\\
19
\end{array}
\right]$,
encode as $b$, where $b = G\cdot m$.
Decode: receive $b$. If there were no errors in transmission, we need to solve $G\cdot m = b$ for $m$. We know how to solve these kinds of systems! Detect Errors: receive $b$. Compute $H\cdot b$. If there were no errors, this would give the zero vector. If there are errors, but in no more than two of the five positions, you are guaranteed that $H\cdot b \neq 0$ |
| Input matrix | Operations | Output matrix |
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