New Constructions of Binary Sequences With Optimal Autocorrelation Value/Magnitude

• Published in: IEEE transactions on information theory Vol. 56; no. 3; pp. 1278 - 1286
• Main Authors: Tang, Xiaohu, Gong, Guang
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.03.2010; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Almost difference set; Applied sciences; Arrays; Autocorrelation; Binary sequences; Complement; Construction; Context; Correlation analysis; Cryptography; Equivalence; Exact sciences and technology; Gongs; Hardware; Information processing; Information theory; Information, signal and communications theory; interleaved sequences; Mathematical functions; Mobile communication; Multiaccess communication; optimal autocorrelation magnitude; optimal autocorrelation value; Optimization; Radar; System testing; Telecommunications and information theory; Binary sequence; optimal autocorrelation value; interleaved sequences; optimal autocorrelation magnitude; Difference set; binary sequences; Autocorrelation; Almost difference set; Legendre polynomial
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2009.2039159

In this paper, we give three new constructions of binary sequences of period AN with optimal autocorrelation value or optimal autocorrelation magnitude using N × 4 interleaved sequences. Yu and Gong recently found any binary sequence of period AN with optimal autocorrelation value constructed from an almost difference set by Arasu et al. is an N × 4 interleaved sequence for which all four columns in its N × 4 array are shift equivalent up to the complement. We found that it is not necessary that four columns are shift equivalent. Instead, it could be a pair of related sequences together with their shifts as the column sequences. The first construction is to use a generalized GMW sequence of period N = 2 k - 1 and its modified version, the second construction is to use a twin prime sequence of length N = p(p + 2) and its modified version, and the third construction, a pair of Legendre sequences of period N = p (p odd prime) with their respective first terms complementary (the 2-level autocorrelation property is not needed for the Legendre sequence). The comparison with the known constructions are given. For the new sequences with optimal autocorrelation value, their corresponding new almost difference sets are also derived.