Optical orthogonal codes: Their bounds and new optimal constructions

A (v, k, λa, λc) optical orthogonal code C is a family of (0, 1)-sequences of length v and weight k satisfying the following two correlation properties: (1) ∑ 0≤t≤v−1xtxt+i ≤ λa for any x = (x0, x1, . . . , xv−1) and any integer i 6≡ 0 mod v; and (2) ∑ 0≤t≤v−1xtyt+i ≤ λb for any x = (x0, x1, . . . , xv−1), y = (y0, y1, . . . , yv−1) with x 6= y, and any integer i, where subscripts are taken modulo v. The study of optical orthogonal codes is motivated by an application in optical code-division multiple-access communication systems. In this article, upper bounds on the size of an optical orthogonal code are discussed. Several new constructions for optimal (v, k, 1, 1) optical orthogonal codes are described by means of optimal cyclic packing families. Many new optimal optical orthogonal codes with weight k ≥ 4 and correlation constraints λa = λc = 1 are thus produced.