Semiframes with Block Size Three and Odd Group Size

Abstract

A (k,λ)‐semiframe of type gu is a (k,λ)‐GDD of type gu, (X,G,B), in which the collection of blocks scriptB can be written as a disjoint union B=P∪Q where scriptQ is partitioned into parallel classes of scriptX and scriptP is partitioned into holey parallel classes, each holey parallel class being a partition of scriptX∖Gj for some Gj∈G. A (k,λ)‐SF(p,d,gu) is a (k,λ)‐semiframe of type gu in which there are p parallel classes in scriptQ and d holey parallel classes with respect to Gj∈G. In this paper, we shall show that there exists a (3, 1)‐SF(p,d,gu) for any g≡3(mod6) if and only if u≥5, u≡1(mod2), p≡u−12(prefixmod0.28emu−1), d=g2−pu−1, and (p,d,g,u)≠(2,1,3,5).

Completing the spectrum of semiframes with block size three

“…Thus by Lemma 2.2 we only need to solve the existence of a 3‐SF for and . From [3] for each there exists a 3‐SF which is also a 3‐RGDD of type , and a 3‐SF whose parallel classes form a 2‐balanced set. Give each point weight 4 or 12.…”

On the existence of k $k$‐cycle semiframes for even k $k$

“…In the following direct constructions, we will use “base blocks with adders” method to obtain 1‐balanced sets. Similar method can be found in [5].…”

“…For our proof in the next sections we limit $$H={K}_{k}$$ or $${C}_{k}$$ in the following constructions of this section. Since the proofs of the following three constructions are similar to the constructions in [5, 7] we just present them and omit their proofs.…”

“…Theorem 1.4 (Alspach et al [1], Cao [4], Cao et al [5,7], Kageyama and Miao [10], Rees [16], and Stinson [19]).…”