On the maximum order complexity of the Thue-Morse and Rudin-Shapiro sequence

Expansion complexity and maximum order complexity are both finer measures of pseudorandomness than the linear complexity which is the most prominent quality measure for cryptographic sequences. The expected value of the $N$th maximum order complexity is of order of magnitude $\log N$ whereas it is easy to find families of sequences with $N$th expansion complexity exponential in $\log N$. This might lead to the conjecture that the maximum order complexity is a finer measure than the expansion complexity. However, in this paper we provide two examples, the Thue-Morse sequence and the Rudin-Shapiro sequence with very small expansion complexity but very large maximum order complexity. More precisely, we prove explicit formulas for their $N$th maximum order complexity which are both of largest possible order of magnitude $N$. We present the result on the Rudin-Shapiro sequence in a more general form as a formula for the maximum order complexity of certain pattern sequences.