Splitting Balance Incomplete Block Designs

In the investigation of authentication codes Ogata, Kurosawa, Stinson and Saido [W. Ogata, K. Kurosawa, D.R. Stinson and H. Saido, New combinatorial designs and their applications to authentication codes and secret sharing schemes, Discrete Math., 2004, 279, 383-405] found that splitting balanced incomplete block designs can be used to construct k-splitting A-codes, whose impersonation attack probabilities and substitution attack probabilities all achieve their information-theoretic lower bounds. There has been some work done on the existence of splitting balanced incomplete block designs: Du [B. Du, Splitting balanced incomplete block designs, The Australasian Journal of Combinatorics, 2005, 31, 287-298, B. Du, Splitting balanced incomplete block designs with block size 3 × 2, J. Combin. Designs, 2004, 12, 404-420] and Ge, Miao and Wang [G. Ge, Y. Miao, L. Wang, Combinatorial constructions for optimal splitting authentication codes, SIAM Journal on Discrete Mathematics, 2005, 18, 663-678] gave the spectra of {v, b, l = u×k, A)-splitting BIBDs for (u, k) = (2,2), (2,3), (3,2) and (4,2); Wang [J. Wang, A new class of optimal 3-splitting authentication codes, Designs, Codes and Cryptography, 2006, 38, 373-381] gave the spectra of (v, b,l = u × k, λ)-splitting BIBDs for (u, k) = (3,3). This article investigates the existence of splitting balanced incomplete block designs (v, 2 × k, λ)-splitting BIBDs, and gives the spectrum of (v, 2 × k, λ)-splitting BIBDs for k = 4,5. We also give some constructions of resolvable splitting BIBDs.