Combinatorial constructions for maximum optical orthogonal signature pattern codes

An (m, n, k, λa, λc) optical orthogonal signature pattern code (OOSPC) is a family C of m × n (0, 1)-matrices of Hamming weight k satisfying two correlation properties. OOSPCs find application in transmitting two-dimensional image through multicore fiber in CDMA networks. Let Θ(m, n, k, λa, λc) denote the largest possible number of codewords among all (m, n, k, λa, λc)-OOSPCs. An (m, n, k, λa, λc)-OOSPCwithΘ(m, n, k, λa, λc) codewords is said to be maximum. For the case λa = λc = λ, the notations (m, n, k, λa, λc)-OOSPC and Θ(m, n, k, λa, λc) can be briefly written as (m, n, k, λ)-OOSPC and Θ(m, n, k, λ). In this paper, some direct constructions for (3, n, 4, 1)-OOSPCs, which are based on skew starters and an application of the Theorem of Weil on multiplicative character sums, are given for some positive integer n. Several recursive constructions for (m, n, k, 1)-OOSPCs are presented by means of incomplete different matrices and group divisible designs. By utilizing those constructions, the number of the codewords of a maximum (m, n, 4, 1)OOSPC is determined for any positive integers m, n such that gcd(m, 18) = 3 and n ≡ 0 (mod 12). It is established that Θ(m, n, 4, 1) = (mn − 12)/12 for any positive integers m, n such that gcd(m, 18) = 3 and n ≡ 0 (mod 12). © 2013 Elsevier B.V. All rights reserved.