Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgments
- Historical Introduction
- 1 General Properties of Correlation
- 2 Applications of Correlation to the Communication of Information
- 3 Finite Fields
- 4 Feedback Shift Register Sequences
- 5 Randomness Measurements and m-Sequences
- 6 Transforms of Sequences and Functions
- 7 Cyclic Difference Sets and Binary Sequences with Two-Level Autocorrelation
- 8 Cyclic Hadamard Sequences, Part 1
- 9 Cyclic Hadamard Sequences, Part 2
- 10 Signal Sets with Low Crosscorrelation
- 11 Correlation of Boolean Functions
- 12 Applications to Radar, Sonar, Synchronization, and CDMA
- Bibliography
- Index
8 - Cyclic Hadamard Sequences, Part 1
Published online by Cambridge University Press: 15 August 2009
- Frontmatter
- Contents
- Preface
- Acknowledgments
- Historical Introduction
- 1 General Properties of Correlation
- 2 Applications of Correlation to the Communication of Information
- 3 Finite Fields
- 4 Feedback Shift Register Sequences
- 5 Randomness Measurements and m-Sequences
- 6 Transforms of Sequences and Functions
- 7 Cyclic Difference Sets and Binary Sequences with Two-Level Autocorrelation
- 8 Cyclic Hadamard Sequences, Part 1
- 9 Cyclic Hadamard Sequences, Part 2
- 10 Signal Sets with Low Crosscorrelation
- 11 Correlation of Boolean Functions
- 12 Applications to Radar, Sonar, Synchronization, and CDMA
- Bibliography
- Index
Summary
Binary sequences of period N with 2-level autocorrelation have many important applications in communications and cryptology. From Section 7.1, 2-level autocorrelation sequences are in natural correspondence with cyclic Hadamard difference sets with ν = N, κ = (N − 1)/2, and λ = (N − 3)/4. For this reason, they are named cyclic Hadamard sequences. In this chapter, 2-level autocorrelation always means ideal 2-level autocorrelation. There are three classic constructions for binary 2-level autocorrelation sequences that were known before 1997 (including some generalizations along these lines after 1997). One is m-sequences, described in Chapter 5, with period N = 2n − 1. The second construction is based on a number theory approach, including three types of sequences in Chapter 2, which are the quadratic residue sequences, Hall sextic residue sequences, and twin prime sequences. The period of such a sequence is either a prime or a product of twin primes. The third construction is associated with intermediate subfields. The resulting sequences have subfield decompositions and period N = 2n − 1. They include GMW sequences, cascaded GMW sequences, and generalized GMW sequences. Although the resulting sequences are binary, this construction relies heavily on intermediate fields and compositions of functions. As a consequence, it involves sequences over intermediate fields that are not binary sequences. The content of this chapter is organized as follows.
- Type
- Chapter
- Information
- Signal Design for Good CorrelationFor Wireless Communication, Cryptography, and Radar, pp. 219 - 266Publisher: Cambridge University PressPrint publication year: 2005