ca)

Write down the matrix whose rows are the given codewords:

$$\var{codeword_matrix(codewords)}$$

Now reduce this matrix to row echelon form. That means the first non-zero entry in each row is further right than in the row above it and that entry is a 1 and there are only zeros above and below that entry. Remove any rows made up of all zeros.

From the matrix above, we obtain:

$$\mathrm{G} = \var{codeword_matrix(g)}$$

b)

A generator matrix for a code is a standard generator matrix if it is of the form $(\mathrm{I}_{k}|\mathrm{A})$, that is, the $k \times k$ identity matrix attached to an arbitrary matrix $\mathrm{A}$.

If a code has a standard generator matrix, the row reduced echelon form will be it.

$\mathrm{G}$ is not of the required form, so it is not a standard generator matrix.

c)

To encode a data vector $\mathbf{v}$, multiply it on the right by the generator matrix for the code.

So we have:

$$\begin{align} \simplify[rowvector]{{vector(data_vectors[0])}*{codeword_matrix(g)}} &= \var[rowvector]{vector(encoded_data_vectors[0])} \\[0.5em] \simplify[rowvector]{{vector(data_vectors[1])}*{codeword_matrix(g)}} &= \var[rowvector]{vector(encoded_data_vectors[1])} \\[0.5em] \simplify[rowvector]{{vector(data_vectors[2])}*{codeword_matrix(g)}} &= \var[rowvector]{vector(encoded_data_vectors[2])} \\[0.5em] \end{align}$$

That is,

$$\begin{align} \var{data_vectors[0]} &\mapsto \var{encoded_data_vectors[0]} \\[1em] \var{data_vectors[1]} &\mapsto \var{encoded_data_vectors[1]} \\[1em] \var{data_vectors[2]} &\mapsto \var{encoded_data_vectors[2]} \\[1em] \end{align}$$

d)

We are looking for a data vector of length $\var{len(g)}$ which is encoded as $c = \var{to_decode}$.

Note that, since $\mathrm{G}$ is in reduced echelon form, the leading non-zero entry in each row is $1$, so the elements of the data vector are the digits of $c$ corresponding to each of the columns of $G$ containing a leading non-zero entry. The leading $1$s in $\mathrm{G}$ are in columns 1, 3, 4, 5. The corresponding digits of $c$ give the data vector $\var{decoded_word}$.

We can confirm that this vector is encoded as $c$:

$$\simplify[rowvector]{ {vector(decoded_word)}*{codeword_matrix(g)} } = \var[rowvector]{vector(to_decode)}$$