Asymptotic behavior of normalized linear complexity of ultimately nonperiodic binary sequences

• Published in: IEEE transactions on information theory Vol. 50; no. 11; pp. 2911 - 2915
• Main Authors: Zongduo Dai, Shaoquan Jiang, Imamura, K., Guang Gong
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.11.2004; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Applied sciences; Asymptotic properties; Binary sequences; Complexity; Computer science; Construction; Cryptography; Exact sciences and technology; Galois fields; Information security; Information systems; Information theory; Information, signal and communications theory; Intervals; Linear feedback shift registers; Signal and communications theory; Telecommunications and information theory; Linear complexity; Binary sequence; Stream ciphering; Cryptography; Asymptotic behavior
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2004.836704

For an ultimately nonperiodic binary sequence s={s/sub t/}/sub t/spl ges/0/, it is shown that the set of the accumulation values of the normalized linear complexity, L/sub s/(n)/n, is a closed interval centered at 1/2, where L/sub s/(n) is the linear complexity of the length n prefix s/sup n/=(s/sub 0/,s/sub 1/,...,s/sub n-1/) of the sequence s. It was known that the limit value of the normalized linear complexity is equal to 0 or 1/2 if it exists. A method is also given for constructing a sequence to have the closed interval [1/2-/spl Delta/, 1/2+/spl Delta/](0/spl les//spl Delta//spl les/1/2) as the set of the accumulation values of its normalized linear complexity.