区组长为4的自反有向平衡不完全区组设计

Existence of Self-Converse Directed BIBDs with Block Size Four

如果从一个有向平衡不完全区组设计DB(k,λ;v)(X,B)到(X,B-1)之间存在一个同构映射f,则这个DB(k,λ;v)被称为自反的,记为SCDB(k,λ;v)(X,B,f),其中B-1={B-1:B∈B},当B=(x1,x2,…,xk-1,xk)时B-1=(xk,xk-1,…,x2,x1).本文主要证明了SCDB(4,λ;v)存在的充分必要条件是λ≡1,2(mod3)时,v≡1(mod3)且v≥4,(v,λ)≠(7,1);λ≡0(mod3)时,v为≥4的任意整数.

A directed balanced incomplete block design DB(k,λ;v) (X,B) is called self_converse if there is an isomorphic mapping f from (X,B) to (X,B-1),where B-1={B-1:B∈B}, and B-1=(xk,xk-1,...,x2,x1) for B=(x1,x2,...,xk-1,xk).In this paper, we show that SCDB(4,λ;v) exists if and only if v≡1 (mod 3) and v≥4,(v,λ)≠(7,1) when λ≡1,2(mod 3); v≥4 when λ≡0(mod 3).