Optimal Two‐Dimensional Optical Orthogonal Codes with the Best Cross‐Correlation Constraint

Abstract

The study of optical orthogonal codes has been motivated by an application in an optical code‐division multiple access system. From a practical point of view, compared to one‐dimensional optical orthogonal codes, two‐dimensional optical orthogonal codes tend to require smaller code length. On the other hand, in some circumstances only with good cross‐correlation one can deal with both synchronization and user identification. These motivate the study of two‐dimensional optical orthogonal codes with better cross‐correlation than auto‐correlation. This paper focuses on optimal two‐dimensional optical orthogonal codes with the auto‐correlation λa and the best cross‐correlation 1. By examining the structures of w‐cyclic group divisible designs and semi‐cyclic incomplete holey group divisible designs, we present new combinatorial constructions for two‐dimensional (n×m,k,λa,1)‐optical orthogonal codes. When k=3 and λa=2, the exact number of codewords of an optimal two‐dimensional (n×m,3,2,1)‐optical orthogonal code is determined for any positive integers n and m≡2(mod4).

Optimal $2$-D $(n\times m,3,2,1)$-optical orthogonal codes and related equi-difference conflict avoiding codes

“…where (a, b) ∈ {(0, 12), (2,6), (3,8), (4,14), (5,5), (6,7), (11,13), (1,16), (14,22), (16,19), (18,24), (7,28), (12,25), (13,27), (22,29), (8,30), (9,32), (10,34), (20,31), (24,33), (15,35), (17,36), (19,37), (21,38), (23,39), (33,15), (34,…”

“…However, there are few results on optimal 2-D (n × m, k, 2, 1)-OOCs when n = 1 in the literature. The only known results for k = 3 is from [17,40], which determined the size of an optimal 2-D (n × m, 3, 2, 1)-OOCs with m ≡ 2 (mod 4). This paper continues the work in [17], and we are concerned about optimal 2-D (n × m, 3, 2, 1)-OOCs with m ≡ 0 (mod 4).…”

Some progress on optimal $ 2 $-D $ (n\times m,3,2,1) $-optical orthogonal codes

“…When n = 1, there are few results on optimal 2-D (n × m, k, 2, 1)-OOCs in the literature. For k = 3, the size of an optimal 2-D (n × m, 3, 2, 1)-OOC with m ≡ 2 (mod 4) was determined thoroughly in [18,40]. Recently, several optimal 2-D (n × m, 3, 2, 1)-OOCs with m ≡ 0 (mod 4) were constructed (see Theorem 2 of [19]).…”

New CRT sequence sets for a collision channel without feedback

“…We remark that deterministic MAC protocols can also be referred to in the literature as conflict-avoiding codes (CACs) [21,22,23,24,25], optical orthogonal codes (OOCs) [26,27,28], or topological transparent scheduling [29,30,31,32] with different design goals. In particular, UI sequences aim to minimize the sequence period by assuming all users are active, whereas CACs aim to maximize the number of potential users when the sequence period and the number of maximum active users are both given.…”