# Maximal likelihood Error/Syndrome table for $[16, 11]$ hamming code

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?

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 $$c=y-z$$.