Existence of strong difference families and constructions for eight new 2‐designs

Abstract

Using group rings and characters as in the theory of abelian difference sets, some nonexistence results for strong difference families are provided. Existences of strong difference families with base block size 3 ≤ k ≤ 9 are discussed. Via strong difference families, eight 2‐ ( v , k , λ ) designs whose existences were unknown are constructed for ( v , k , λ ) ∈ { ( 3417 , 8 , 1 ) , ( 3753 , 8 , 1 ) , ( 2665 , 9 , 1 ) , ( 3817 , 9 , 1 ) , ( 4393 , 9 , 1 ) , ( 7353 , 9 , 1 ) , ( 12537 , 9 , 1 ) , ( 345 , 9 , 3 ) }.

On balanced $(Z_{4u}\times Z_{8v},\{4,5\},1)$ difference packings

“…The concept of a strong difference family was introduced in [5] to provide constructions for relative difference families (see also [6], [10], [26]). Some work also had been done on strong difference families recently [11], [15], [16], [33]. In [2], a strong difference family was used to construct 3-pyramidal KTSs (Kirkman triple systems).…”

“…Appendix A (u, v, g, h) =(2,36,2,4) {(0,0), (1,14), (1,15), (1,26)}, {(0,0),(0,5),(0,15),(0, 19),(1,7)}, {(0,0),(0,2),(0,8),(1,13)}, {(0,0),(0,3), (1,4), (1,20), (1,33)}.…”