On the structure of optimal error-correcting codes

Kabatyanskii and Panchenko asked whether two sets of size 10 consisting of binary 7-tuples exist, such that all 100 sums with one element from each set are distinct. This question is here answered in the negative by showing that the existence of such sets would imply the existence of a binary single-error-correcting code of length 9 and size 40 (which is unique) with a certain property, which such a code does not have.