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### Solution: #### Part a) Which of the following binary codes are linear? A binary code \(\mathcal{C}\) is linear if: 1. The zero vector is in \(\mathcal{C}\). 2. The sum of any two codewords in \(\mathcal{C}\) is also in \(\mathcal{C}\) (closed under addition). --- i) \(\mathcal{C} = \{(0, 0, 0, 0), (1, 0, 1, 0), (0, 1, 1, 0), (1, 1, 1, 0)\}\) - Zero vector: \((0, 0, 0, 0) \in \mathcal{C}\). ✔️ - Closure under addition: - \((1, 0, 1, 0) + (0, 1, 1, 0) = (1, 1, 0, 0) \notin \mathcal{C}\). Since the sum of \((1, 0, 1, 0)\) and \((0, 1, 1, 0)\) is not in \(\mathcal{C}\), the code is not linear. Answer for i): Not linear. --- ii) \(\mathcal{C} = \{(0, 0, 0), (1, 1, 0), (1, 0, 1), (0, 1, 1)\}\) - Zero vector: \((0, 0, 0) \in \mathcal{C}\). ✔️ - Closure under addition: - \((1, 1, 0) + (1, 0, 1) = (0, 1, 1) \in \mathcal{C}\). - \((1, 1, 0) + (0, 1, 1) = (1, 0, 1) \in \mathcal{C}\). - \((1, 0, 1) + (0, 1, 1) = (1, 1, 0) \in \mathcal{C}\). All sums _____ _________ _____ ______ ________ ______ _______ ____ ________ _________.
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