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An updated review on cross-correlation of m-sequences

Tor Helleseth, Chunlei Li

TL;DR

This work surveys the crosscorrelation properties of maximum-length m-sequences over finite fields, focusing on decimations that produce spectra with a small number of distinct values. It highlights the deep connections between crosscorrelation, Walsh transforms of trace-power functions, and cyclic-code weight distributions, employing algebraic and number-theoretic tools such as exponential sums and Kloosterman-type sums. The chapter enumerates eleven infinite three-valued decimation families and numerous four-, five-, and six-valued cases (including Niho-type exponents), along with key open problems that guide ongoing research. The findings have practical impact on sequence design for CDMA, GPS-like systems, and cryptographic applications by enabling construction of sequences with low crosscorrelation and favorable nonlinear properties.

Abstract

Maximum-length sequences (m-sequences for short) over finite fields are generated by linear feedback shift registers with primitive characteristic polynomials. These sequences have nice mathematical structures and good randomness properties that are favorable in practical applications. During the past five decades, the crosscorrelation between m-sequences of the same period has been intensively studied, and a particular research focus has been on investigating the cross-correlation spectra with few possibles values. In this chapter we summarize all known results on this topic in the literature and promote several open problems for future research.

An updated review on cross-correlation of m-sequences

TL;DR

This work surveys the crosscorrelation properties of maximum-length m-sequences over finite fields, focusing on decimations that produce spectra with a small number of distinct values. It highlights the deep connections between crosscorrelation, Walsh transforms of trace-power functions, and cyclic-code weight distributions, employing algebraic and number-theoretic tools such as exponential sums and Kloosterman-type sums. The chapter enumerates eleven infinite three-valued decimation families and numerous four-, five-, and six-valued cases (including Niho-type exponents), along with key open problems that guide ongoing research. The findings have practical impact on sequence design for CDMA, GPS-like systems, and cryptographic applications by enabling construction of sequences with low crosscorrelation and favorable nonlinear properties.

Abstract

Maximum-length sequences (m-sequences for short) over finite fields are generated by linear feedback shift registers with primitive characteristic polynomials. These sequences have nice mathematical structures and good randomness properties that are favorable in practical applications. During the past five decades, the crosscorrelation between m-sequences of the same period has been intensively studied, and a particular research focus has been on investigating the cross-correlation spectra with few possibles values. In this chapter we summarize all known results on this topic in the literature and promote several open problems for future research.
Paper Structure (7 sections, 25 theorems, 68 equations)

This paper contains 7 sections, 25 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. Crosscorrelation of m-sequences
  3. Three-valued crosscorrelation
  4. Four-valued crosscorrelation
  5. Five-valued crosscorrelation
  6. Six-valued crosscorrelation
  7. Conclusion

Key Result

Lemma 1

Let $f(x) \in \mathbb{F}_{p}[x]$ be a primitive polynomial of degree $n$ and $\{s_t\}$ be an m-sequence over $\mathbb{F}_{p}$ generated by the linear recurrence associated with $f(x)$. Then the sequence $\{s_t\}$ satisfies the following properties: In particular, in the case of $p=2$, the binary m-sequence $\{s_t\}$ also satisfies:

Theorems & Definitions (30)

  • Lemma 1
  • Definition 1: Crosscorrelation between m-sequences
  • Lemma 2
  • Definition 2: Walsh Transform
  • Definition 3
  • Lemma 3
  • Remark 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • ...and 20 more