Splitting Directed Balanced 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 theirapplications to authentication codes and secret shares schemes, Discrete Math, 2004, 279,383-405]found that splitting balanced incomplete block designs can be used to constructk-splitting A-codes, whose impersonation attack probabilities and substitution attackprobabilities all achieve their information-theoretic lower bounds. There has been somework 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, 404-420land Ge, Miao and Wang [G. Ge, Y. Miao, L. Wang,Combinatorial constructions for optimal splitting authentication codes, SIAM Journal onDiscrete Mathematics, 2005, 18, 663-678]gave the spectra of (v,b,l= 2×k,λ)-splittingBIBDs for (u,k)=(2,e),(2,3),(3,2)and(4,2); Wang[J. Wang, A new class of optimal3-splitting authentication codes, Designs, codes and Cryptography, 2006, 38, 373-381]gavethe spectra of (v,b,l=2×k,λ)-splitting BIBDs for (u, k)=(3, 3).This article investigates the existence of splitting directed balanced incomplete blockdesigns(v, l=2×k,λ), and gives the spectrum of (v, l= 2×k,λ)-splitting DBIBDs fork=4.