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Let K = { k 1 , k 2 , … , k r } and L = { l 1 , l 2 , … , l s } be disjoint subsets of { 0 , 1 , … , p − 1 } , where p is a prime and A = { A 1 , A 2 , … , A m } is a family of subsets of [ n ] = { 1 , 2 , … , n } such that | A i | ( mod p ) ∈ K for all A i ∈ A and | A i ∩ A j | ( mod p ) ∈ L for i ≠ j . In 1991, Alon, Babai and Suzuki conjectured that if n ≥ s + max 1 ≤ i ≤ r k i , then | A | ≤ n s + n s − 1 + ⋯ + n s − r + 1 . In 2000, Qian and Ray-Chaudhuri proved the conjecture under the condition n ≥ 2 s − r . In 2015, Hwang and Kim verified this conjecture. In this paper, we will prove that if n ≥ 2 s − 2 r + 1 or n ≥ s + max 1 ≤ i ≤ r k i , then | A | ≤ n − 1 s + n − 1 s − 1 + ⋯ + n − 1 s − 2 r + 1 . This result strengthens both the upper bound of Alon–Babai–Suzuki’s conjecture and Qian and Ray-Chaudhuri’s result, when n ≥ 2 s − 2 . [ABSTRACT FROM AUTHOR]

A uniformly resolvable design (URD) is a resolvable design in which each parallel class contains blocks of only one block size k . Such a class is denoted k -pc and for a given k the number of k -pcs is denoted r k . The number of points of the URD is denoted by v . In the literature, the existence of URDs of block sizes k 1 and k 2 with { k 1 , k 2 } ∈ { { 2 , 3 } , { 2 , 4 } , { 3 , 4 } } has been studied with much efforts. For { k 1 , k 2 } = { 2 , 3 } or { 3 , 4 } , the existence spectra have been determined completely, while for { k 1 , k 2 } = { 2 , 4 } there are still 1338 undetermined pairs of ( v , r 2 ) . In this paper we continue the study on the existence of URDs of block sizes 2 and 4. We improve the known results by reducing the number of open cases from 1338 to 16 . In addition, we prove that there exist 4 -RGDDs of types 2 178 , 2 250 and 2 334 . [ABSTRACT FROM AUTHOR]

A uniformly resolvable design (URD) is a resolvable design in which each parallel class contains blocks of only one block size k . Such a class is denoted k -pc and for a given k the number of k -pcs is denoted r k . Let v denote the number of points of the URD. For the case of block sizes 3 and 4 (both existing), the necessary conditions imply that v ≡ 0 ( mod 12 ) . It has been shown that almost all URDs with permissible r 3 and r 4 exist for v ≡ 0 ( mod 24 ) , v ≡ 0 ( mod 60 ) , v ≡ 36 ( mod 144 ) or v ≡ 36 ( mod 108 ) . In this paper, we prove that the necessary conditions for the existence of a URD with block sizes 3 and 4 are also sufficient, except when v = 12 , r 3 = 1 and r 4 = 3 . [ABSTRACT FROM AUTHOR]

The sizes of optimal constant-composition codes of weight three have been determined by Chee, Ge and Ling with four cases in doubt. Group divisible codes played an important role in their constructions. In this paper, we study the problem of constructing optimal ternary constant-composition codes with Hamming weight four and minimum distance six. The problem is solved with a small number of lengths undetermined. The previously known results are those with code length no greater than 10. [ABSTRACT FROM AUTHOR]

In this paper, we continue to investigate the existence of 5 -GDDs, 4-frames and 4-RGDDs, which have been studied by numerous researchers in the past two decades due to their vital applications in the constructions for other types of designs. Much progress has been made for the existence problems of these designs, while the problems are still open. Our new results mainly lie in the cases with relatively large group size. We also construct some smaller designs, including 4 -RGDDs of types 2 184 , 9 44 , 14 10 , 18 18 , 22 10 and 36 11 . [ABSTRACT FROM AUTHOR]

Abstract: Non-uniform group divisible designs (GDDs) and non-uniform Kirkman frames are useful in the constructions for other types of designs. In this paper, we consider the existence problems for -GDDs of type with and Kirkman frames of type . First, we determine completely the spectrum for uniform -GDDs of type . Then, we consider the entire existence problem for non-uniform -GDDs of type with . We show that, for each given , up to a small number of undetermined cases of , the necessary conditions on for the existence of a -GDD of type are also sufficient, except possibly when for and when for . Finally, a similar result for Kirkman frames of type is obtained. We show that, for each given , up to a small number of undetermined cases of , the necessary conditions on for the existence of a Kirkman frame of type are also sufficient, except possibly when for and when for . [Copyright &y& Elsevier]

Abstract: Non-uniform group divisible designs are instrumental in the constructions for other types of designs. Most of the progress for the existence of -GDDs of type is on the case when is even, where the existence for small has played a key role. In order to determine the spectrum for -GDDs of type with being odd, we continue to investigate the small cases with in this paper. We show that, for each , the necessary conditions for the existence of a -GDD of type are also sufficient. As the applications of these GDDs, we obtain a few pairwise balanced designs with minimum block size 4. Meanwhile, we also improve the existence result for frame self-orthogonal Mendelsohn triple systems of type by reducing an infinite class of possible exceptions, namely and , to eight undetermined cases. [Copyright &y& Elsevier]