Splitting Balanced Incomplete Block Designs with Block Size 2×k

IntroductionIn the study of authentication codes,Ogata,etal[1] ,introduced the concept of splitting balancedincomplete block design which can be used toconstructk- splitting A- codes.A ( v,u× k,λ) - splitting balanced incompleteblock design is a pair( V,B) ,where V is a v- setofpoints and Bis a collection of subses of V ( calledblocks) with size uk such that the followingconditions are satisfied:( 1 ) Every B∈ B is expressed as a disjointunion of u subblocks of size k:B=B1∪ B2 ∪ ...…     

On the Existence of （v, 3 × 3,λ）-Splitting Balanced Incomplete Block Design with A between 2 to 9

Let (υ, u × c, λ)-splitting BIBD denote a (υ, u × c, λ)-splitting balanced incomplete block design of order υ with block size u × c and index λ. The necessary conditions for the existence of a (υ, u × c, λ)-splitting BIBD are υ ⩾ uc, λ(υ − 1) ≡ 0 0 mod (c(u − 1)) and λυ(υ − 1) ≡ 0 mod (c ² u(u − 1)). In this paper, for 2 ⩽ λ ⩽ 9 the necessary conditions for the existence of a (υ, 3 × 3, λ)-splitting BIBD are also sufficient with one possible exception for (υ, λ) = (39, 9). Foundation item: the National Natural Science Foundation of China (No. 10771193); the Starter Foundation for the Doctors of Zhejiang Gongshang University (No. 1020XJ030517); the Natural Science Foundation of Universities of Jiangsu Province (No. 07KJB110090); the Starter Foundation for the Doctors of Nantong University (No. 07B12)