Directed Pairwise Balanced Design With Block Sizes From K={4,8}

A directed pairwise balanced design,of order v with block sizes from K and index 1,denoted by DB(K,1;v),is a pair(X,β) where X is a set of vpoints andβis a collection of transitively ordered subsets(called blocks) of X with block sizes from K, such that every ordered pair of distinct points of X occurs in precisely one block ofβ.Directed pairwise balanced designs are closely related to t-deletion/insertion-correcting codes.R.Fuji-Hara,Y.Miao,J.Wang and J.Yin have determined the spectra of DB(K,1;v) when K={4,5} and K={4,6}.In this paper,we consider the existence of directed pairwise balanced design with block sizes from K={4,8}.We prove that DB({4,8},1;v) dose not exist when v∈{5,6,9,11,12,14,15,17,18,21,23}; and take use of Construction of direct and recursive structure of the method to show that there exists a DB({4,8},1;v) for any other positive integer v≥4,(except for v∈{24,26,27,33,35,39,45,51,62,74,102,114,126,138,186}).