On Bicovers of Pairs by Quintuples: $v$ odd, $v\equiv 3(mod10)$

Abstract

A bicover of pairs by quintuples of a $v$-set $V$ is a family of 5-subsets of $V$ (called blocks) with the property that every pair of distinct elements from $V$ occurs in at least two blocks. If no other such bicover has fewer blocks, the bicover is said to be minimum, and the number of blocks in a minimum bicover is the covering number $C_2(v,5,2)$, or simply $C_2(v)$. It is well known that $C_2(v)\ge \lceil \frac{v\lceil (v-1)/2\rceil }{5}\rceil =B_2(v)$, where $\lceil x\rceil$ is the least integer not less than $x$. It is shown here that if $v$ is odd and $v\not\equiv 3$ mod 10, $v\ne 9$ or 15,then $C_2(v)=B(v)$.