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 D with point set X , block set B and index λ is called completely reducible super-simple (CRSS), if its block set B can be written as B = λ i=1 B i , such that B i forms the block set of a design with index unity but having the same parameters as D for each 1 ≤ i ≤ λ. It is easy to see, the existence of CRSS designs with index λ implies that of CRSS designs with index i for 1 ≤ i ≤ λ. 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) ∈ {(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 ∈ {4, 5, 6, 9}.

Decomposable super‐simple RBIBDs with block size 4 and index 6

“…For decomposable super‐simple designs, not much is known except for completely reducible super‐simple designs and decomposable super‐simple ‐NRBIBDs . Motivated by these, in this paper, we investigate the existence of ‐decomposable super‐simple ‐RBIBDs.…”

“…(0, 9,13,25), (20,22,24,27), (4, 10, 12, 30), (3,7,23,26), (2,5,14,19), (8,21,28,29), (1,11,17,18), ∞ (6, 15, 16, ). v = 36: (0, 9, 10, 13), (1,11,28,32), (12,15,23,34), (7,8,19,25), (2,4,16,31), (14,20,21,30), (17,24,26,29), ( Proof. These designs are over Table 3 displays a suitable initial parallel class of the first super-simple (4, 3) sub-RGDD of type g 5 .…”

Decomposable super‐simple NRBIBDs with block size 4 and index 6

“…In , the existence of super‐simple ‐NRBIBDs (with a few extra whist properties) was investigated, and solutions were obtained for (mod 4), , and . For decomposable super‐simple designs, not much is known except for the completely reducible super‐simple designs . Motivated by these, in this paper, we investigate the existence of 3‐decomposable super‐simple ‐NRBIBDs.…”

“…≥ v 17, and v ∉ {17, 21, 117, 129, 141, 145, 165, 177, 185, 189, 209, 213}. For decomposable super-simple designs, not much is known except for the completely reducible super-simple designs [7,33,34]. Motivated by these, in this paper, we investigate the existence of 3-decomposable super-simple v ( , 4, 6)-NRBIBDs.…”

A pair of disjoint 3-GDDs of type g t u 1

“…In [7], the concept of group divisible design is generalized to a new code named group divisible code, which is shown useful in recursive constructions for constant-weight and constant-composition codes. One can also find applications of disjoint group divisible designs in the determination of more optimal constant-weight codes (see, for example, [19,20]). …”