Jingyuan Chen

A design is said to be super‐simple if any two of its blocks have at most two common points. A design with index is said to be ‐ decomposable , if its blocks can be partitioned into nonempty collections , such that each with the point set forms a design with index . In this paper, it is proved that there exists a ‐DSS ‐GDD (decomposable super‐simple group divisible design) of type if and only if and , and except possibly for .

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 .

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 ‐decomposable, if its blocks can be partitioned into nonempty collections , , such that each with the point set forms a design with index . In this paper, it is proved that there exists a ‐decomposable super‐simple ‐BIBD (balanced incomplete block design) if and only if and .

A design is said to be super‐simple if the intersection of any two of its blocks has at most two elements. A design with index is said to be ‐decomposable, if its blocks can be partitioned into nonempty collections , , such that each with the point set forms a design with index . In this paper, it is proved that for , there exists a ‐decomposable super‐simple ‐GDD of type if and only if , and , except for , and except possibly for .

Let be a positive integer, be a magic square, where . is called most perfect magic square (MPMS for short) if , and . Let , where . is called rational if both and possess the property that the sums of the numbers in every row and every column are the same; otherwise, is said to be irrational. It was shown that there exists an MPMS if and only if and . In this paper, it is proved that there exists an irrational MPMS if and only if and .

Decomposable designs are important combinatorial designs, and have applications in constant weight codes, perfect threshold schemes and graph decompositions. Let be an even positive integer. A ‐decomposable super‐simple ‐balanced incomplete block design (BIBD) is a super‐simple ‐BIBD, , where , and each is a ‐BIBDs for . In this paper, for , it is proved that the necessary conditions for the existence of a ‐decomposable super‐simple ‐BIBD are also sufficient.