 Three-error syndrome decoding of extended Golay code

For a binary symmetric matrix $A$ with $A^2=I$, the syndrome of an error $(p,q)$ relative to the column parity-check matrix $(I;A)$ is $s=p+qA$. An error of total Hamming weight at most three has at most one nonzero bit in one half. Test $s$, each $s+A_i$, $sA$, and each $sA+A_i$ for weights at most three, two, three and two respectively. These identify $(s,0)$, $(s+A_i,e_i)$, $(0,sA)$ and $(e_i,sA+A_i)$. For the extended binary Golay code, minimum Hamming distance eight guarantees that at most one such error exists.