Linear complexity profile and correlation measure of interleaved sequences

Maximum-Order Complexity and Correlation Measures

“…Now we return to the case m = 2. Even if the correlation measure of order k is large for some small k, we may be still able to derive a nontrivial lower bound on the maximum-order complexity by substituting the correlation measure of order k by its analog with bounded lags, see [7] for the analog of (1). For example, the two-prime generator T = (t i ) ∞ i=0 , see [3], of length pq with two odd primes p < q satisfies t i + t i+p + t i+q + t i+p+q = 0 if gcd(i, pq) = 1 and its correlation measure of order 4 is obviously close to pq, see [16].…”

On the Nth linear complexity of automatic sequences

“…However, its linear complexity profile is quite large, see [3]. This lower bound on the linear complexity profile can be obtained using an analog of the result of [4], see [14], for a modified correlation measure with bounded lags. Examples of sequences with large correlation measure of small order with bounded lags but large linear complexity profile were not known before.…”