On Ternary Complementary Pairs

Let A = {a0, ... aℓ-1 }, B = {b0, . . . , bℓ-1 } be two finite sequences of length ℓ. Their nonperiodic autocorrelation function NA,B (s) is defined as: NA,B(s) = ∑aiai+1 + ∑b1bi+1, s = 0,…,ℓ-1 where x* is the complex conjugate of x. If NA,B (s) = 0 for s = 1, ... , ℓ - 1 then A, B is called a complementary pair. If, furthermore, ai, bi ε {-1,1}, i = 0,...,ℓ-1 , or, ai, bi ε {-1,0,1}, i = 0, ... , ℓ-1, then A, B is called a binary complementary pair (BCP), or, a ternary complementary pair (TCP), respectively. A BCP is also called Golay sequences. A TCP is a generalisation of a BCP. Since Golay sequences are only known to exist for lengths n = 2a10b26c, a, b, c ≥ 0, recent papers have focused on TCP's. The purpose of this paper is to give an overview of existing constructions and techniques and present a variety of new constuctions, new restrictions on the deficiences and new computational results for TCP's. In particular: We give new constructions which concatenate shorter group of sequences to obtain longer sequences. Many of these constructions can be applied recursively and lead to infinite families of TCP's. We give many new restrictions on TCP's of lengths ℓ and deficiencies ∂ = 2x, where x ≡ ℓmod 4. We settle all the cases for existence/non existence of TCP's of lengths ℓ ≤ 20 and weights w ≤ 40. We give TCP's with minimum deficiencies for all lengths ℓ ≤ 22.