1. **State the problem:** Given a generator matrix $G$ and a message vector $\mathbf{m} = (m_3, m_2, m_1, m_0) = (1,0,1,0)$, find the code vector $\mathbf{c}$. 2. **Formula used:** The code vector is found by multiplying the message vector by the generator matrix: $$\mathbf{c} = \mathbf{m} \times G$$ 3. **Important rules:** - The multiplication is done in modulo 2 arithmetic (binary addition and multiplication). - Each element of the resulting vector is computed by summing the products of corresponding elements modulo 2. 4. **Intermediate work:** - Write the message vector as a row vector: $\mathbf{m} = [1 \ 0 \ 1 \ 0]$ - Multiply by $G$ (assuming $G$ is known; since $G$ is not provided, the general step is shown): $$\mathbf{c} = [1 \ 0 \ 1 \ 0] \times G$$ - Compute each element of $\mathbf{c}$ by summing the products modulo 2. 5. **Explanation:** - Each bit of the code vector is a linear combination of the message bits weighted by the generator matrix columns. - This process encodes the message into a codeword suitable for error detection or correction. **Final answer:** The code vector $\mathbf{c}$ is obtained by multiplying $\mathbf{m}$ by $G$ modulo 2: $$\mathbf{c} = (1,0,1,0) \times G$$ Since $G$ is not specified, the exact code vector cannot be computed here.