Statistical Span Property of Binary Run Sequences

We define run sequences of period 2^{<italic>n</italic>} - 1 as the binary sequences where the distribution of runs of 0&#x2019;s and runs of 1&#x2019;s is exactly same as that for the maximal length linear shift resister sequences of period 2^{<italic>n</italic>} - 1. We first count the number of all the cyclically distinct run sequences of period 2^{<italic>n</italic>} - 1. For each <italic>n</italic>-tuple, we consider the average number of occurrences over all the run sequences of period 2^{<italic>n</italic>} - 1. We identify the <italic>n</italic>-tuples with average number 1 and, in particular, those that occur exactly once in every run sequence of period 2^{<italic>n</italic>} - 1. We finally prove that, as <italic>n</italic> increases, the average number of every non-zero <italic>n</italic>-tuple approaches to 1.