Optimal 2-D $(n\times m,3,2,1)$-optical Orthogonal Codes

Optical orthogonal codes are commonly used as signature codes for optical code-division multiple access systems. So far, research on 2-D optical orthogonal codes has mainly concentrated on the same autocorrelation and cross-correlation constraints. In this paper, we are concerned about optimal 2-D optical orthogonal codes with the autocorrelation λ
a
and the cross-correlation 1. Some combinatorial constructions for 2-D (
n
×
m
,
k
,λ
a
,1) -optical orthogonal codes are presented. When
k
=3 and λ
a
=2, the exact number of codewords of an optimal 2-D (
n
×
m
,3,2,1)-optical orthogonal code is determined for any positive integers
n
≡ 0,1,3,6,9,10 (mod 12) and
m
≡ 2(mod 4).