Optimal (4up, 5, 1) optical orthogonal codes

Abstract

By a ðv v; k; 1Þ-OOC we mean an optical orthogonal code. In this paper, it is proved that an optimal ð4p; 5; 1Þ-OOC exists for prime p 1 ðmod 10Þ, and that an optimal ð4up; 5; 1Þ-OOC exists for u ¼ 2; 3 and prime p 11 ðmod 20Þ. These results are obtained by applying Weil's theorem. #

New results on optimal (v, 4, 2, 1) optical orthogonal codes

“…Optical orthogonal codes have great importance in the applications [19] and for a long time the ones most investigated have been the optimal (v, k, 1)-OOCs [9,18] with special attention to those that are difference families [1,8,13,15,16,26] or relative difference families [11,14,21] in view of their importance in design theory [2,5,6,10]. In the last years several interesting results have also been obtained about optimal (v, k, λ)-OOCs with λ = 2 [3,17,20] and even with λ > 2 [4].…”

Strong difference families, difference covers, and their applications for relative difference families

“…The case when = 3 and q is a prime in the above proposition was shown by Buratti [9], which was generalized to any by Chang and Ji in [13] (also see [11]). We denote the set of all prime powers q satisfying r q ( , e)/e ≥ t by P(e, , t).…”

“…In fact, many infinite series of optimal (v, k, 1)-OOCs were obtained from cyclic relative difference families. For some known results, we refer to [9,[12][13][14][18][19][20]25,33] and references there in.…”

Semi-cyclic holey group divisible designs with block size three

“…We have that if p satisfies p − 5 √ p − 12 > 0, which yields p > 45, then there exists an element x ∈ Z p such that x ∈ C 2 1 , x + 1 ∈ C 2 1 and x − 1 ∈ C 2 0 . When 5 ≤ p ≤ 43, it is readily checked that we may take x as (p, x) = (5, 2), (7,5), (11,6), (13,5), (17,5), (19,2), (23,10), (29,2), (31,11), (37,5), (41, 6), (43,2). ✷ Lemma 6.9 There exists a 3-SCHGDD of type (6, 2 p ) for any prime p ≥ 3.…”