[math-fun] A binary sieve

Choose an integer k >= 2 , let q' = 2^k - 1 , initialise set S = {1, ..., q'-1} ; then delete every j for which i < j exists with j == i*2^h (mod q') for some h . For example, when k = 5 , q' = 31 , finally S = {1, 3, 5, 7, 11, 15} . I conjecture that max S = (q'-1)/2 for all k ; but why? Fred Lunnon