# Optimal Resolvable Steiner Quintuple System covering with circles and ellipses

Here is a resolvable Steiner quintuple system. Every tuple from 1-25 appears in exactly one of the sets.
{{1,2,3,4,5},{6,7,8,9,10},{11,12,13,14,15},{16,17,18,19,20},{21,22,23,24,25},
{1,6,11,16,21},{2,7,12,17,22},{3,8,13,18,23},{4,9,14,19,24},{5,10,15,20,25},
{1,7,13,19,25},{2,8,14,20,21},{3,9,15,16,22},{4,10,11,17,23},{5,6,12,18,24},
{1,10,14,18,22},{2,6,15,19,23},{3,7,11,20,24},{4,8,12,16,25},{5,9,13,17,21},
{4,6,13,20,22},{25,2,9,11,18},{16,23,5,7,14},{12,19,21,3,10},{8,15,17,24,1},
{4,18,7,21,15},{6,25,14,3,17},{13,2,16,10,24},{20,9,23,12,1},{22,11,5,19,8}}

It's possible to cover this as a point system with ellipses and circles. An optimal covering would be one where each ellipse and circle clearly went through exactly 5 points, and didn't come close to any others. Is there a good way to optimize this?

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This incidence structure is not a Steiner quintuple system (they can only exist when $v=3,5\pmod{6}$). What you have is a $2$-design, and in particular a $(25,5,1)$-BIBD.