We determine the possible even sets of nodes on sextic surfaces in ${\mathbb{P}}^3$, showing in particular that their cardinalities are exactly the numbers in the set $\{24,32,40,56\}$. We also show that all the possible cases admit an explicit description. The methods that we use are an interplay of coding theory and projective geometry on one hand, and of homological and computer algebra on the other. We give a detailed geometric construction for the new case of an even set of 56 nodes, but the ultimate verification of existence...