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Golay Complementary Sequences of Arbitrary Length and Asymptotic Existence of Hadamard Matrices

Cheng Du, Yi Jiang

TL;DR

This work addresses the challenge of constructing Golay complementary sequence (GCS) sets of arbitrary length with small cardinality and advancing Hadamard matrix existence. It introduces a multiplicative-additive construction that yields a 4-phase GCS octet of cardinality $2^{3+\lceil \log_2 r\rceil}$ for any length $n$, where $r$ is the number of nonzero digits in the base-$10^{13}$ expansion of $n$, and proves that octets cover all lengths up to $10^{13}$. By leveraging the matrix representation of the signed symmetric group and complex base sequences, the paper also builds Hadamard matrices from GCS in novel ways and derives an improved asymptotic bound: there exist block-circulant Hadamard matrices of order $2^t m$ for any odd $m$, with $t = 6\lfloor \frac{1}{40}\log_2 m\rfloor + 10$. The contributions collectively broaden the practical reach of GCS in communications and tighten the asymptotic landscape for Hadamard existence, offering new pathways to Hadamard constructions at large scales. The work suggests further reductions in the exponent by refining Hadamard-from-GCS methods and exploring deeper CBS-based compositions.

Abstract

In this work, we construct $4$-phase Golay complementary sequence (GCS) set of cardinality $2^{3+\lceil \log_2 r \rceil}$ with arbitrary sequence length $n$, where the $10^{13}$-base expansion of $n$ has $r$ nonzero digits. Specifically, the GCS octets (eight sequences) cover all the lengths no greater than $10^{13}$. Besides, based on the representation theory of signed symmetric group, we construct Hadamard matrices from some special GCS to improve their asymptotic existence: there exist Hadamard matrices of order $2^t m$ for any odd number $m$, where $t = 6\lfloor \frac{1}{40}\log_{2}m\rfloor + 10$.

Golay Complementary Sequences of Arbitrary Length and Asymptotic Existence of Hadamard Matrices

TL;DR

This work addresses the challenge of constructing Golay complementary sequence (GCS) sets of arbitrary length with small cardinality and advancing Hadamard matrix existence. It introduces a multiplicative-additive construction that yields a 4-phase GCS octet of cardinality for any length , where is the number of nonzero digits in the base- expansion of , and proves that octets cover all lengths up to . By leveraging the matrix representation of the signed symmetric group and complex base sequences, the paper also builds Hadamard matrices from GCS in novel ways and derives an improved asymptotic bound: there exist block-circulant Hadamard matrices of order for any odd , with . The contributions collectively broaden the practical reach of GCS in communications and tighten the asymptotic landscape for Hadamard existence, offering new pathways to Hadamard constructions at large scales. The work suggests further reductions in the exponent by refining Hadamard-from-GCS methods and exploring deeper CBS-based compositions.

Abstract

In this work, we construct -phase Golay complementary sequence (GCS) set of cardinality with arbitrary sequence length , where the -base expansion of has nonzero digits. Specifically, the GCS octets (eight sequences) cover all the lengths no greater than . Besides, based on the representation theory of signed symmetric group, we construct Hadamard matrices from some special GCS to improve their asymptotic existence: there exist Hadamard matrices of order for any odd number , where .
Paper Structure (14 sections, 18 theorems, 76 equations, 3 figures, 2 tables)

This paper contains 14 sections, 18 theorems, 76 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Golay Complementary Sequence Set
  4. Perfect Sequences over Signed Symmetric Group
  5. Construction of GCS of Arbitrary Length
  6. Feasible Lengths of GCS Sets Constructed from GCS pairs
  7. Denser Existence from Complex Base Sequences
  8. Construction of 4-phase GCS of Arbitrary Length
  9. Construction of Hadamard Matrices from GCS
  10. Hadamard Matrix from Perfect Sequence
  11. Sequences Combining
  12. Periodic Complementary Sequences with Quasi-symmetric and Supplementary Structure
  13. Asymptotic Existence of Hadamard Matrices
  14. Conclusions

Key Result

Proposition 1

Given a nontrivial 2-phase GCS pair $\{{\bf a}, {\bf b}\}$ with sequence length $s$, two polyphase GCS pairs $\{{\bf c}, {\bf d}\}$ and $\{{\bf e}, {\bf f}\}$ with sequence lengths $t$ and $u$ respectively, and the recursive compositions the so-obtained $\{{\bf g}, {\bf h}\}$ is a polyphase GCS pair of length $stu$.

Figures (3)

  • Figure 1: Densities of $4$-phase GCS sets.
  • Figure 2: Hierarchical applications of Theorem \ref{['thm:compromise']} in $\lceil \log_2 r \rceil$ levels.
  • Figure 3: Asymptotic Existence of Hadamard Matrices. (Note that $m$ is odd.)

Theorems & Definitions (36)

  • Definition 1
  • Proposition 1: craigen2002complex
  • Corollary 1: craigen2002complex
  • Corollary 2
  • Remark 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Theorem 1
  • Lemma 1
  • ...and 26 more