The structure and enumeration of periodic binary sequences with high nonlinear complexity

Nonlinear complexity, as an important measure for assessing the randomness of sequences, is defined as the length of the shortest feedback shift registers that can generate a given sequence. In this paper, the structure of n-periodic binary sequences with nonlinear complexity larger than or equal to 3n/4 is characterized. Based on their structure, an exact enumeration formula for the number of such periodic sequences is determined.

💡 Research Summary

The paper investigates binary sequences of period n whose nonlinear complexity (nlc) – defined as the length of the shortest nonlinear feedback shift register (NFSR) that can generate the sequence – is at least ⌊3n/4⌋. While linear complexity has been extensively studied, the behavior of nlc for periodic sequences below the near‑maximum values (n‑1, n‑2) remains largely unexplored. The authors fill this gap by providing a complete structural description of such high‑nlc periodic sequences and by deriving an exact enumeration formula for their number.

The authors first recall the definition of nonlinear complexity introduced by Jansen and Boecke and the basic lemma that nlc equals one plus the length of the longest pair of identical subsequences occurring at least twice with different successors. They then introduce the family B(n,c,d) of aperiodic finite‑length sequences parameterized by a target complexity c (≥⌊n/2⌋) and a spacing parameter d (1 ≤ d ≤ min{n‑c,⌊n/2⌋}). When d = n‑c, the sequence is forced to be aperiodic regardless of the choice of the trailing part s