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

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 = 4 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> − 1 is a prime. (2) v is a product of twin primes. (3) v = 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> − 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 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> − 1 for n ≤ 10 are listed and classified according to known construction methods.