Completely reducible super‐simple (v,4,4) $(v,4,4)$‐BIBDs and related constant weight codes

Abstract A design is said to be super‐simple if the intersection of any two blocks has at most two elements. A design with index is said to be completely reducible , if its blocks can be partitioned into nonempty collections , such that each together with the point set forms a design with index unity. In this paper, it is proved that there exists a completely reducible super‐simple (CRSS) balanced incomplete block design (‐BIBD for short) if and only if and or . A ‐ary constant weight code (CWC) of length with weight and distance is denoted as a code. The maximum size of a code is denoted as , and the codes achieving this size are called optimal . CRSS designs with index are closely related to ‐ary CWCs. By using the results of CRSS ‐BIBDs, s are determined for all .