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I think I have to start with a parity check matrix for $[16,11]$ Hamming code. $$H = \left( \begin{array}{cccccccccccccccc} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 \\ 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 \\ 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ \end{array} \right)$$

How do I go about finding the syndrome decoding table?

If I understand it correctly, my syndrome table will contain all the possible permutations of 0s and 1s upto $2^{5}$. E.g. $00000,...,01111,...,11111$. Do I have to find out coset leaders for all 32 syndromes?.

If yes, how will the decoding work?

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Indeed, you need to find coset leaders for all cosets of the code in ${\Bbb F}_2^{16}$. So if a word $y$ is received, the syndrome $s=Hy\in{\Bbb F}_2^5$ is computed. Suppose $z\in{\Bbb F}_2^{16}$ is the corresponding coset leader, then the decoding gives the codeword $c=y-z$.

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