= 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.
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