Two-dimensional optical orthogonal codes and semicyclic group divisible designs

• Authors: Jianmin Wang;Jianxing Yin
• Venue: IEEE Transactions on Information Theory
• Year: 2010

A (n × m, k, ρ) two-dimensional optical orthogonal code (2-D OOC), C, is a family of n × m(0,1)-arrays of constant weight k such that Σi=1n Σj=0m-1 A(i,j)B(i,j ⊕m τ) ≤ ρ for any arrays A, B in C and any integer ρ except when A = B and τ ≡ 0 (mod m), where ⊕m denotes addition modulo m. Such codes are of current practical interest as they enable optical communication at lower chip rate. To simplify practical implementation, the AM-OPPW (at most one-pulse per wavelength) restriction is often appended to a 2-D OOC. An AM-OPPW 2-D OOC is optimal if its size is the largest possible. In this paper, the notion of a perfect AM-OPPW 2-DOOCis proposed, which is an optimal (n×m, k, ρ) AM-OPPW 2-DOOCwith cardinality mρ n(n-1)...(n-ρ)/k(k-1)...(k-ρ). A link between optimal (n × m, k, ρ) AM-OPPW 2-D OOCs and block designs is developed. Some new constructions for such optimal codes are described by means of semicyclic group divisible designs. Several new infinite families of perfect (n × m, k, 1) AM-OPPW 2-D OOCs with k ∈ {2, 3, 4} are thus produced.