On existence of Two Classes of Generalized Howell Designs with Block Size Three and Index Two

Abstract

Let $$t, k, {\lambda }, s$$ and v be nonnegative integers, and let X be a set of v symbols. A generalized Howell design, denoted t-GHD $$_k (s, v; {\lambda })$$ , is an $$s \times s$$ array, each cell of which is either empty or contains a k-set of symbols from X, called a block, such that: (i) each symbol appears exactly once in each row and in each column (i.e. each row and column is a resolution of X); (ii) no t-subset of elements from X appears in more than $${\lambda }$$ cells. A generalized Howell design is a class of doubly resolvable designs , which generalize a number of well-known objects. Particular instances of the parameters correspond to generalized Howell designs are doubly resolvable group divisible designs (DRGDDs). In this paper, we concentrate on the case that $$t=2,k=3$$ and $${\lambda }= 2$$ , and simply write GHD(s, v; 2). The spectrum of GHD $$(3n-3,3n;2)$$ ’s and GHD $$(6n-6,6n;2)$$ ’s is completely established by solving the existence of (3, 2)-DRGDDs of types $$3^n$$ and $$6^n$$ . At the same time, we also survey rummage the existence of GHD $$_4(n,4n;1)$$ ’s. As their applications, several new classes of multiply constant-weight codes are obtained.