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 $\lambda$ is said to be completely reducible, if its blocks can be partitioned into nonempty collections $B_i,1\le i\le \lambda$, such that each $B_i$ 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) $\left(v,4,4\right)$ balanced incomplete block design ($\left(v,4,4\right)$-BIBD for short) if and only if $v\ge 13$ and $v\equiv 1$ or $4\left(\mathrm{mod}12\right)$. A $q$-ary constant weight code (CWC) of length $v$ with weight $w$ and distance $d$ is denoted as a ${\left(v,d,w\right)}_q$ code. The maximum size of a ${\left(v,d,w\right)}_q$ code is denoted as $A_q\left(v,d,w\right)$, and the ${\left(v,d,w\right)}_q$ codes achieving this size are called optimal. CRSS designs with index $q-1$ are closely related to $q$-ary CWCs. By using the results of CRSS $\left(v,4,4\right)$-BIBDs, $A_5\left(v,6,4\right)$s are determined for all $v\equiv 0,1,3,4\left(\mathrm{mod}12\right),v\ge 12$.