On Bicovers of Pairs by Quintuples: \(v\) odd, \(v ≡ 3 (mod 10)\)

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 \(C_2(v) \geq \left \lceil \frac{v\left \lceil{(v-1)/2}\right \rceil}{5} \right \rceil = B_2(v)\), 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, \(v\not=9\) or 15,then \(C_2(v)=B(v)\).