Completely reducible super-simple designs with block size five and index two
Complete reducible super-simple (CRSS) designs are closely related to \(q\)-ary constant weight codes. A \((v,k,\lambda )\)-CRSS design is just an optimal \((v,2(k-1),k)_{\lambda +1}\) code. In this paper, we mainly investigate the existence of a \((v,5,2)\)-CRSS design and show that such a design exists if and only if \(v\equiv 1,5\pmod {20}\) and \(v\ge 21\), except possibly when \(v = 25\). Using this result, we determine the maximum size of an \((n,8,5)_3\) code for all \(n\equiv 0,1,4,5 \pmod {20}\) with the only length \(n=25\) unsettled. In addition, we also construct super-simple \((v,5,3)\)-BIBDs for \(v=45,65\).
completely reducible super-simple designs super-simple designs constant weight codes group divisible designs primary 05b05 94b25