Word Count : 549

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We are asked to find the parity-check matrix (\overline{\mathcal{H}}_3) and decode several vectors. Let’s solve it manually step by step. --- Assume (\overline{\mathcal{H}}_3) corresponds to a standard ((8,4)) Hamming code (since vectors are length 8), then the parity-check matrix is formed from all non-zero 3-bit binary columns: [ \overline{\mathcal{H}}_3 = \begin{bmatrix} 1 & 0 & 0 & 1 & 0 & 1 & 1 & 0 \ 0 & 1 & 0 & 1 & 1 & 0 & 1 & 0 \ 0 & 0 & 1 & 0 & 1 & 1 & 1 & 0 \end{bmatrix}. ] But to make it easier, let's denote Hamming (8,4) ________ ____ ____ ____ ________ __________ ____ ________ ___ _____ _______.
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