Existence of cyclic Hadamard difference sets and its relation to binary sequences with ideal autocorrelation

Jeong Heon Kim, Hong-Yeop Song

Research output: Contribution to journalArticle

15 Citations (Scopus)

Abstract

Balanced binary sequences with ideal autocorrelation are equivalent to (v, k, λ)-cyclic Hadamard difference sets with v = 4n - 1, k = 2n - 1, λ = n - 1 for some positive integer n. Every known cyclic Hadamard difference set has one of the following three types of v : (1) v = 4n - 1 is a prime. (2) v is a product of twin primes. (3) v = 2n - 1 for n = 2, 3, ⋯. It is conjectured that all cyclic Hadamard difference sets have parameter v which falls into one of the three types. The conjecture has been previously confirmed for n < 10000 except for 17 cases not fully investigated. In this paper, four smallest cases among these 17 cases are examined and the conjecture is confirmed for all v ≤ 3435. In addition, all the inequivalent cyclic Hadamard difference sets with v = 2n - 1 for n ≤ 10 are listed and classified according to known construction methods.

Original languageEnglish
Pages (from-to)14-18
Number of pages5
JournalJournal of Communications and Networks
Volume1
Issue number1
DOIs
Publication statusPublished - 1999 Jan 1

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Binary sequences
Autocorrelation

All Science Journal Classification (ASJC) codes

  • Information Systems
  • Computer Networks and Communications

Cite this

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abstract = "Balanced binary sequences with ideal autocorrelation are equivalent to (v, k, λ)-cyclic Hadamard difference sets with v = 4n - 1, k = 2n - 1, λ = n - 1 for some positive integer n. Every known cyclic Hadamard difference set has one of the following three types of v : (1) v = 4n - 1 is a prime. (2) v is a product of twin primes. (3) v = 2n - 1 for n = 2, 3, ⋯. It is conjectured that all cyclic Hadamard difference sets have parameter v which falls into one of the three types. The conjecture has been previously confirmed for n < 10000 except for 17 cases not fully investigated. In this paper, four smallest cases among these 17 cases are examined and the conjecture is confirmed for all v ≤ 3435. In addition, all the inequivalent cyclic Hadamard difference sets with v = 2n - 1 for n ≤ 10 are listed and classified according to known construction methods.",
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Existence of cyclic Hadamard difference sets and its relation to binary sequences with ideal autocorrelation. / Kim, Jeong Heon; Song, Hong-Yeop.

In: Journal of Communications and Networks, Vol. 1, No. 1, 01.01.1999, p. 14-18.

Research output: Contribution to journalArticle

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