Abstract :
An ( m , n , k , λ a , λ c ) 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 , λ a , λ c ) denote the largest possible number of codewords among all ( m , n , k , λ a , λ c ) -OOSPCs. An ( m , n , k , λ a , λ c ) -OOSPC with Θ ( m , n , k , λ a , λ c ) codewords is said to be maximum. For the case λ a = λ c = λ , the notations ( m , n , k , λ a , λ c ) -OOSPC and Θ ( m , n , k , λ a , λ c ) 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 ) = ( m n − 12 ) / 12 for any positive integers m , n such that gcd ( m , 18 ) = 3 and n ≡ 0 ( mod 12 ) .
