Two-Dimensional Optical Orthogonal Codes and Semicyclic Group Divisible Designs

A (n x m , k , ¿) two-dimensional optical orthogonal code (2-D OOC), C , is a family of n x m (0, 1)-arrays of constant weight k such that ¿i =1 n ¿ j =0 m -1 A ( i , j ) B ( i , j ¿ m ¿) ¿ ¿ for any arrays A , B in C and any integer ¿ except when A = B and ¿ ¿ 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-D OOC is proposed, which is an optimal (n x m , k , ¿) AM-OPPW 2-D OOC with cardinality [(m ¿ n ( n -1)...( n -¿))/( k ( k -1)...( k -¿))] . A link between optimal (n x m , k , ¿) 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 x m , k , 1) AM-OPPW 2-D OOCs with k ¿ {2, 3, 4} are thus produced.