Discussion
The number of code word possible using the generator \(G = \begin{bmatrix} 1 & 1 & 0 \\ 0 & 1 & 1 \end{bmatrix}\) is :
Answer: C
4 code words are possible.
A generator matrix with k rows produces every linear combination of its rows over GF(2), so a code with k independent rows has \(2^{k}\) code words. Here G has 2 independent rows of length 3:
\[ \begin{aligned} 0 \cdot r_1 + 0 \cdot r_2 &= 000 \\ r_1 &= 110 \\ r_2 &= 011 \\ r_1 + r_2 &= 101 \end{aligned} \]
That gives \(2^{2} = 4\) code words. 8 would be \(2^{3}\), every 3-bit word, which would need three independent rows. 1 and 2 forget either the all-zero word or the sum of the two rows, both of which are always in a linear code.
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