A code is said to be a w-identifiable parent property code (or w-IPP code for short) if whenever d is a descendant of w (or fewer) codewords, and one can always identify at least one of the parents of d. Let C be an (N,w + 1,q)-code and C* an (w + 1)-color graph for C. If a graph G is a subgraph of C* and consists of w +1 edges with different colors, then G is called a (w +1)-pattern of C*. In this paper, we proved that C is a w-IPP code if and only if there exists at most one vertex with color degree more than 1 in any (w + 1)-pattern of C*.
This paper investigated the existence of splitting balanced incomplete block designs with blck size 2× k .The necessary conditions for such a design are λ(v-1) ≡0(mod k ),and λv(v- 1)≡0(mod 2 k 2).It will show that the above necessary conditions are also sufficient for k =3 with the definite exception( v,λ )=(10,1) and with several possible exceptions.
The necessary and sufficient conditions for the existence of simple incomplete block design （v, ω; 4, 2）-IPBDs are determined. As a consequence, the necessary and sufficient conditions for the embeddings of simple two-fold balanced incomplete block designs with block size 4 are also determined.
