The index of a binary word

A binary word u is f -free if it does not contain f as a factor. A word f is d-good if for any f -free words u and v of length d, v can be obtained from u by complementing one by one the bits of u on which u and v differ, such that all intermediate words are f -free. We say that f is good if it is d-good for any d ≥ 1. A word is bad if it is not good. The index β(f) of f is the smallest integer d such that f is not d-good, so that β(f) < ∞ if and only if f is bad. It is proved that β(f) < |f | holds for any bad word f . In addition, β(f) < 2|f | holds for almost all bad words f and it is conjectured that the same holds for all bad words. An infinite family of words such that each member of it is bad, but 2-good, is constructed. It is conjectured that the words of this family are all the words that are bad and 2-good among those with exactly two 1s. These conjectures are supported by computer experiments.