Completely reducible super-simple designs with block size four and related super-simple packings

Abstract

A design is said to be super-simple if the intersection of any two blocks has at most two elements. A super-simple design $${\mathcal{D}}$$ with point set X, block set $${\mathcal{B}}$$ and index ? is called completely reducible super-simple (CRSS), if its block set $${\mathcal{B}}$$ can be written as $${\mathcal{B}=\bigcup_{i=1}^{\lambda} \mathcal{B}_i}$$ , such that $${\mathcal{B}_i}$$ forms the block set of a design with index unity but having the same parameters as $${\mathcal{D}}$$ for each 1 ? i ? ?. It is easy to see, the existence of CRSS designs with index ? implies that of CRSS designs with index i for 1 ? i ? ?. CRSS designs are closely related to q-ary constant weight codes (CWCs). A (v, 4, q)-CRSS design is just an optimal (v, 6, 4) q+1 code. On the other hand, CRSS group divisible designs (CRSSGDDs) can be used to construct q-ary group divisible codes (GDCs), which have been proved useful in the constructions of q-ary CWCs. In this paper, we mainly investigate the existence of CRSS designs. Three neat results are obtained as follows. Firstly, we determine completely the spectrum for a (v, 4, 3)-CRSS design. As a consequence, a class of new optimal (v, 6, 4)4 codes is obtained. Secondly, we give a general construction for (4, 4)-CRSSGDDs with skew Room frames, and prove that the necessary conditions for the existence of a (4, 2)-CRSSGDD of type g u are also sufficient except definitely for $${(g,u)\in \{(2,4),(3,4),(6,4)\}}$$ . Finally, we consider the related optimal super-simple (v, 4, 2)-packings and show that such designs exist for all v ? 4 except definitely for $${v\in \{4,5,6,9\}}$$ .