Coset Leaders and Syndromes

Question

This is my Parity check matrix For Coset Leader 100010 Syndrome is 101 Could any one help me with the procedure, since I figured out that adding up both the columns isn't much of help.

1 1 1 1 0 0
1 1 0 0 1 0
1 0 0 0 0 1

Answer 1

The syndrome of a vector $\vec{v}$ is given by $\vec{v}H^T$. In your case this amounts to the transpose of the sum of the first and fifth columns of $H$ (presumably over $\mathbb{Z}_2$)

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Syndrome of $\vec{v}$ is given by

$$\text{syn}(\vec{v})=\vec{v}H^T=\begin{pmatrix} 1&0&0&0&1&0 \end{pmatrix} \begin{pmatrix}1&1&1\\ 1&1&0\\ 1&0&0\\ 1&0&0\\0&1&0\\ 0&0&1 \end{pmatrix}=\begin{pmatrix} 1\\0\\1 \end{pmatrix}$$