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

N1 - Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.

PY - 1999/3

Y1 - 1999/3

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 -