Combinatorial constructions for maximum optical orthogonal signature pattern codes

An (m,n,k,λa,λc)(m,n,k,λa,λc) optical orthogonal signature pattern code (OOSPC) is a family CC of m×nm×n   (0,1)(0,1)-matrices of Hamming weight kk 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)Θ(m,n,k,λa,λc) denote the largest possible number of codewords among all (m,n,k,λa,λc)(m,n,k,λa,λc)-OOSPCs. An (m,n,k,λa,λc)(m,n,k,λa,λc)-OOSPC with Θ(m,n,k,λa,λc)Θ(m,n,k,λa,λc) codewords is said to be maximum  . For the case λa=λc=λλa=λc=λ, the notations (m,n,k,λa,λc)(m,n,k,λa,λc)-OOSPC and Θ(m,n,k,λa,λc)Θ(m,n,k,λa,λc) can be briefly written as (m,n,k,λ)(m,n,k,λ)-OOSPC and Θ(m,n,k,λ)Θ(m,n,k,λ). In this paper, some direct constructions for (3,n,4,1)(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 nn. Several recursive constructions for (m,n,k,1)(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)(m,n,4,1)-OOSPC is determined for any positive integers m,nm,n such that gcd(m,18)=3gcd(m,18)=3 and n≡0(mod12). It is established that Θ(m,n,4,1)=(mn−12)/12Θ(m,n,4,1)=(mn−12)/12 for any positive integers m,nm,n such that gcd(m,18)=3gcd(m,18)=3 and n≡0(mod12).