### Further results on the existence of super-simple pairwise balanced designs with block sizes 3 and 4

Guangzhou Chen, Yue Guo, Yong Zhang
2018 Advances in Mathematics of Communications
In statistical planning of experiments, super-simple designs are the ones providing samples with maximum intersection as small as possible. Super-simple pairwise balanced designs are useful in constructing other types of super-simple designs which can be applied to codes and designs. In this paper, the super-simple pairwise balanced designs with block sizes 3 and 4 are investigated and it is proved that the necessary conditions for the existence of a super-simple (v, {3, 4}, λ)-PBD for λ = 7, 9
more » ... λ)-PBD for λ = 7, 9 and λ = 2k, k ≥ 1, are sufficient with seven possible exceptions. In the end, several optical orthogonal codes and superimposed codes are given. 2010 Mathematics Subject Classification: Primary: 05B05, 51E05; Secondary: 62K10, 94C30. Key words and phrases: Super-simple designs, pairwise balanced designs, balanced incomplete block designs, group divisible designs, optical orthogonal code. A design is called cyclic if it admits an automorphism σ of order v = |X |, where X is the point set of the design, which can be identified with The concept of super-simple designs was introduced by Gronau and Mullin  . The existence of super-simple designs is an interesting problem by itself, but there are also useful applications. For example, such designs are used in constructing perfect hash families  and coverings  , in the construction of new designs  and in the construction of superimposed codes  . Super-simple pairwise balanced designs are powerful for the construction of other types of combinatorial structures  (such as super-simple designs). In statistical planning of experiments, supersimple designs are the ones providing samples with maximum intersection as small as possible. There are other useful applications in [4, 22, 29] . Let m denote the smallest integer in K. Define α(K) = gcd{k − 1 : k ∈ K} and β(K) = gcd{k(k − 1) : k ∈ K}. The necessary conditions for the existence of a super-simple (v, K, λ)-PBD are v ≥ λ(m − 2) + 2, λ(v − 1) ≡ 0 (mod α(K)) and λv(v − 1) ≡ 0 (mod β(K)). Dehon  proved that the necessary conditions for a simple (v, 3, λ)-BIBD, listed below, are sufficient. Lemma 1.1. () There exists a simple (v, 3, λ)-BIBD if and only if The necessary conditions for the existence of a super-simple (v, 4, λ)-PBD are v ≥ 2λ + 2, λ(v − 1) ≡ 0 (mod 3) and λv(v − 1) ≡ 0 (mod 12). For the existence of super-simple (v, 4, λ)-BIBDs, the necessary conditions are known to be sufficient for λ ∈ {2 − 6, 8, 9} (see [3, 9, 13, 14, 15, 21, 23, 26, 31, 37]). Gronau and Mullin  solved the case for λ = 2, and the corrected proof appeared in  . The λ = 3 case was solved by Chen  . The λ = 4 case was solved independently by Adams, Bryant, and Khodkar  and Chen  . The case of λ = 5 was solved by Cao, Chen and Wei  . The case of λ = 6 was solved by Chen, Cao and Wei  . The case of λ = 8 was solved by Chen, Sun and Zhang  . The case of λ = 9 was solved by Zhang, Chen and Sun . A survey on super-simple (v, 4, λ)-BIBDs with v ≤ 32 appeared in  . We summarize these known results in the following result. Lemma 1.2. ([3, 9, 13, 14, 15, 21, 23, 26, 31, 37] ) The necessary conditions of a super-simple (v, 4, λ)-BIBD for λ = 2, 3, 4, 5, 6, 8, 9 are sufficient. The necessary conditions for the existence of a super-simple (v, 5, λ)-BIBD are known to be sufficient for λ ∈ {2, 3, 4, 5} (see [1, 16, 17, 18, 25, 36] ). For more results on super-simple designs we refer the reader to [2, 5, 8, 11, 19, 20, 28, 34, 36] and references therein. The existence of (v, {3, 4}, 1)-PBD was stated in [23, 27] . Lemma 1.3. ([23, 27] ) There exists a (v, {3, 4}, 1)-PBD if and only if v ≡ 0, 1 (mod 3) and v ≥ 3. Quite recently, Chen et al studied the existence of a super-simple (v, {3, 4}, λ)-PBD for 2 ≤ λ ≤ 6 and showed the following result. Lemma 1.4. ( ) The necessary conditions of a super-simple (v, {3, 4}, λ)-PBD for 2 ≤ λ ≤ 6 are sufficient except possibly for (v, λ)