I don't know of a standard test specifically designed for this purpose, but here is an idea that may work for you. Count the number of positions in the string which contain 0 ones, 1 one, 2 ones, ..., 6 ones. If the strings are independent, then these numbers will follow a Binomial(n=6, p) distribution, where p is the fraction of ones. Specifically, if L is the length of the strings and is the number of positions with i ones, then
You can then use a contingency table (based on the chi square statistic) to compare the expected counts from this formula with the actual counts from your experiment.