Constructions for optimal (υ, 4, 1) optical orthogonal codes

Determination of Sizes of Optimal Three‐Dimensional Optical Orthogonal Codes of Weight Three with the AM‐OPP Restriction

“…We first deal with the case of (v, w) = (1, 2). When u = 14, by Lemma B.2 in Supporting Information, there exists a 2-cyclic 3-IGDD of type 2 (14,2) , which is also a 3-SCGDP of type 2 14 with J * (14 × 1 × 2, 3, 1) base blocks. When u = 26, by Lemma B.5 in Supporting Information, there exists a 2-cyclic 3-IGDD of type 2 (26,11) .…”

“…1-D OOCs were first suggested by [7] in 1989. Since then there are many researches on 1-D OOCs (see, e.g., [2,3,5,12,14,18,19,26]). 1-D OOCs spread optical pulses only in time domain.…”

Constructions of optimal optical orthogonal codes with weight five

“…It has been showed in [9] that an optimal ðv; 3; 1Þ-OOC exists if and only if v 6 ¼ 6t þ 2 with t 2 or 3 (mod 4). Although there are some partial results (see, e.g., [1,3,4,6,7,9,10,14,15,16,23]), the existence problem for an optimal ðv; 4; 1Þ-OOC is far from settled. The only complete congruence classes of v for which the existence of an optimal ðv; 4; 1Þ-OOC was solved are v 6 ðmod 12Þ due to Ge and Yin [16] and v 0 ðmod 24Þ due to Chang, Fuji-Hara, and Miao [4].…”

“…By using Weil's theorem on character sum estimates, Tang and Yin [22] were able to show that an optimal ð15v; 5; 1Þ-OOC always exists, where v is a product of primes congruent to 1 (mod 4) and greater than 5. On the other hand, by using a combinatorial configuration called skew starter, Chen et al [7] and Ge and Yin [16] constructed a g-regular CPð4; 1; guÞ for any g 2 f6; 18; 24; 72g and any positive integer u such that gcdðu; 150Þ ¼ 1 or 25. Furthermore, by using Weil's theorem and skew starters together and combining other powerful recursive construction, Chang et al [4] have shown that an optimal ðv; 4; 1Þ-OOC exists for any positive integer v 0 ðmod 24Þ.…”

Semicyclic 4-GDDs and related two-dimensional optical orthogonal codes

“…Then there exists a 4-SCGDD of type m n whenever m ≡ 0 (mod 6) and n ≥ 5, except when (m, n) =(6,5), and possibly when n = 5, m ≡ ±6(mod 36) and m ≥ 30.5.2The case m ≡ 2, 4 (mod6) In view of Theorems 5.1 and 5.2, we only need to consider n ≡ 1 (mod 3) and n ≥ 7.Lemma 5.11There exists a 4-SCGDD of type 2 n for any n ≡ 1 (mod 3), n ≥ 7 and n = 10; and there does not exist any 4-SCGDD of type 210 .Proof The result for each stated value of n is given in Proposition 3.3.Lemma 5.12There exists a 4-SCGDD of type 4 n for any n ≡ 1 (mod 3) and n ≥ 10; and there does not exist any 4-SCGDD of type 4 7 .Proof A computer search shows that there is no 4-SCGDD of type 4 7 . For n ∈ {10,13,16, 19,22,25,28,31, 34, 37, 40, 43, 49, 52, 58, 61, 67, 73, 79, 94, 97, 103, 109}, the desired design can be found in Theorem 2.7. Now, we take u = 9 for n ∈ E = {46, 55, 64, 82, 100, 172}; u = 15 for n ∈ F = {76, 106, 166}; or u = 12 for n ∈ H = {169}.…”