New families of normal numbers constructed from existing ones

Given an integer q≥2 , let Aq:={0,1,…,q−1} be the set of base q digits. We say that α=a1…ak , where each ai∈Aq , is a word of length λ(α)=k . Let H(1),H(2),… be a sequence of nonnegative integers and given a sequence of words α1,α2,… , we examine under which conditions the number 0.αH(1)1αH(2)2… is a normal number in base q (where αrj={αj…αj}r times ).