Abstract
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).
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References
Ogata W, Kurosawa K, Stinson D R, et al. New combinatorial designs and their applications to authentication codes and secret sharing schemes[J]. Discrete Math, 2004, 279: 383–405.
Simmons G J. Authentication theory/Decoding theory [J]. Lecture Notes in Computer Science, 1985, 96: 411–431.
Simmons G J. Message authentication with arbitration of transmitter/receiver disputes [J]. Lecture Notes in Computer Science, 1988, 304: 151–165.
Simmons G J. A Cartesian product construction for unconditionally secure authentication codes that permit arbitration [J]. J Cryptology, 1990, 2: 77–104.
Du B. Splitting balanced incomplete block designs with block size 3 × 2 [J]. J Combin Des, 2004, 12: 404–420.
Ge G, Miao Y, Wang L. Combinatorial constructions for authentication codes [J]. SIAM Journal on Discrete Math, 2005, 18(4): 663–678.
Wang J. A new class of 3-splitting authentication codes [J]. Des, Codes and Cryptography, 2006, 38(3): 373–381.
Colbourn C J, Hoffman D G, Rees R. A new class of group divisible designs with block size three [J]. J Combin Theory Ser A, 1992, 59: 73–89.
Ge G, Rees R. On group divisible designs with block size four and group type 6u m 1 [J]. Discrete Math, 2004, 279: 247–265.
Colbourn C J, Dinitz J H. CRC handbook of combinatorial designs[M]. Boca Raton, FL: CRC Press, 1996.
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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)
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Su, Rw., Wang, Jh. On the existence of (υ, 3 × 3, λ)-splitting balanced incomplete block design with λ between 2 to 9. J. Shanghai Jiaotong Univ. (Sci.) 13, 482–486 (2008). https://doi.org/10.1007/s12204-008-0482-0
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DOI: https://doi.org/10.1007/s12204-008-0482-0