Cyclic Hadamard Difference Sets — Constructions and Applications

Abstract

A cyclic (v, k, λ) difference set with parameters v = 4t -1, k = 2t - 1, λ = t - 1 is called a cyclic Hadamard difference set. In all known cases, v = 4t-1 is either a prime, a product of twin primes, or one less than a power of 2. When v is a prime (but not a Mersenne prime), all known examples are obtained by the quadratic residue (Legendre sequence) construction or, in addition, when also v = 4 a ²+27, from Hall’s sextic residue construction. When v is a product of twin primes, all known examples arise from the Jacobi symbol (Stanton-Sprott) construction. However, when v = 2ⁿ -1, in addition to the Singer difference set (PN-sequence) construction and (for composite n > 4) the GMW examples, there are several new constructions, all based on the existence of (GF(2ⁿ). All inequivalent examples of cyclic Hadamard difference sets with v = 2ⁿ - 1 for n ≤ 10 have now been identified, and all belong to known or suspected infinite classes of examples. (There are ten inequivalent examples at n = 10.)

Because of their favorable autocorrelation properties, binary sequences corresponding to cyclic Hadamard difference sets are widely used in communications and radar. Several of these applications are described.