Question
Find the parity check matrix of the code . Decode the following vectors
i)
ii)
iii)
iv)
Answer :
Word Count : 549
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Skip in 10
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) parity-check matrix in standard form: [ \overline{\mathcal{H}}_3 = \begin{bmatrix} 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 \ 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 \ 0 & 0 & 0 & 1 & 1 & 1 & 1 & 0 \end{bmatrix}. ] This works for decoding using syndrome decoding. --- ### Step 1: Compute syndrome The syndrome (S = \overline{\mathcal{H}}_3 \cdot r^T) (mod 2) for each received vector (r). #### i) (r_1 = (1\ 1\ 1\ 0\ 0\ 0\ 0\ 1)) Multiply modulo 2: * Row 1: (1\cdot1 +
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