ISL 1988 Problem 57

$S$ is the set of all sequences $\{a_i| 1 \leq i \leq 7, a_i = 0 \text{ or } 1\}.$ The distance between two elements $\{a_i\}$ and $\{b_i\}$ of $S$ is defined as $$\sum^7_{i = 1} |a_i - b_i|.$$ $T$ is a subset of $S$ in which any two elements have a distance apart greater than or equal to 3. Prove that $T$ contains at most 16 elements. Give an example of such a subset with 16 elements.