(v,3×4,1)-分裂平衡不完全区组设计的存在性

On the Existence of Splitting Balanced Incomplete Block Designs

设(v,u×c,λ)-SBIBD表示一个阶数为v区组大小为u×c指数为λ的分裂平衡不完全区组设计.(v,u×c,λ)-SBIBD存在的必要条件是v≥uc,λ(v-1)≡0(mod c(u-1))和λv(v-1)≡0(mod c2u(u-1)).对分裂平衡不完全区组设计的研究不仅有它的理论意义而且它和设计理论、图论以及分裂认证码有着密切的联系.文章利用差的方法和递推构作方法证明了当v≡1(mod 96)时,存在(v,3×4,1)-分裂平衡不完全区组设计.作为它的应用得到了一类最优4-分裂认证码.

Let（v,u×c,λ）-SBIBD denote a（v,u×c,λ）-splitting balanced incomplete block design of order with block size and index.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 λv（v-1）≡0（mod c2u（u-1））.The investigation for the existence of splitting BIBDs has theoretical sig-nificance and is closely connected with design theory,graph theory and splitting authentication codes.In this paper,applying difference method and recursive construction,we show that there exists a（v,3 × 4,1）-splitting balanced in-complete block design for v≡1（mod 96）.As its consequence,we obtain a new infinite class of optimal 4-splitting au-thentication codes.