Optimal ternary synchronization parameter

Determine the optimal synchronization parameter achievable by ternary synchronization strings over the three-symbol alphabet.

Background

The paper resolves the extremal alphabet-size question by proving that ternary synchronization strings exist at every length for a fixed parameter below one, establishing that three symbols are optimal as a constant alphabet size. It also gives a computer-assisted quantitative construction achieving every parameter greater than 226/227, and in particular parameter 227/228.

The remaining issue is quantitative rather than existential: the best possible synchronization parameter, or equivalently the strongest possible normalized edit-distance gap, for ternary synchronization strings is not determined. Improving the explicit 227/228 construction or proving its optimality would address this open problem.