On the Nth linear complexity of automatic sequences

Abstract

The Nth linear complexity of a sequence is a measure of predictability. Any unpredictable sequence must have large Nth linear complexity. However, in this paper we show that for q-automatic sequences over Fq the converse is not true.We prove that any (not ultimately periodic) q-automatic sequence over Fq has Nth linear complexity of order of magnitude N. For some famous sequences including the Thue–Morse and Rudin–Shapiro sequence we determine the exact values of their Nth linear complexities. These are non-trivial examples of predictable sequences with Nth linear complexity of largest possible order of magnitude.