Further Results on the Existence of Splitting BIBDs and Application to Authentication Codes.

Let ( v, u× c, λ)-splitting BIBD denote a ( v, u× c, λ)-splitting balanced incomplete block design of order v with block size u× c and index λ. Necessary conditions for the existence of a ( v, u× c, λ)-splitting BIBD are v≥ uc, λ( v−1)≡0 (mod c( u−1)) and λ v( v−1)≡0 (mod ( c2 u( u−1))). We show in this paper that the necessary conditions for the existence of a ( v,3×3, λ)-splitting BIBD are also sufficient with possible exceptions when (1) ( v, λ)∈{(55,1),(39,9 k): k=1,2,...}, (2) λ≡0 (mod 54) and v≡0 (mod 2). We also show that there exists a ( v,3×4,1)-splitting BIBD when v≡1 (mod 96). As its application, we obtain a new infinite class of optimal 4-splitting authentication codes. [ABSTRACT FROM AUTHOR]