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 $$v=45,65$$ . [ABSTRACT FROM AUTHOR]