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Consider a simple code $\mathcal{C}$ for error detection and correction. Each codeword in $\mathcal{C}$ consists of $2$ data bits $[d_1, d_0]$ followed by check bits $[c_2, c_1, c_0]$. The check bits are computed as follows: $c_2 = d_1+d_0, \: c_1=d_1$ and $c_0=d_0$, where $'+'$ is a modulo-$2$ addition.

- Write down all the codewords for $\mathcal{C}$
- Determine the minimum Hamming distance between any two distinct codewords of $\mathcal{C}$