### 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 = 2^{n} - 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 = 2^{n} - 1 for n ≤ 10 are listed and classified according to known construction methods.

Original language | English |
---|---|

Pages (from-to) | 14-18 |

Number of pages | 5 |

Journal | Journal of Communications and Networks |

Volume | 1 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1999 Jan 1 |

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### All Science Journal Classification (ASJC) codes

- Information Systems
- Computer Networks and Communications

### Cite this

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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.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Existence of cyclic Hadamard difference sets and its relation to binary sequences with ideal autocorrelation

AU - Kim, Jeong Heon

AU - Song, Hong-Yeop

PY - 1999/1/1

Y1 - 1999/1/1

N2 - 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.

AB - 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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U2 - 10.1109/JCN.1999.6596693

DO - 10.1109/JCN.1999.6596693

M3 - Article

AN - SCOPUS:0002668939

VL - 1

SP - 14

EP - 18

JO - Journal of Communications and Networks

JF - Journal of Communications and Networks

SN - 1229-2370

IS - 1

ER -