Improved Lower Bounds

Lower Bounds (later results)

Theorem 7.11 of [ABGdH07] in the annotated bibliography shows that the following covering designs cannot match the Schonheim bound:
(29,5,2), (41,6,2), (55,7,2), (71,8,2), (89,9,2).
Thanks to Malcolm Greig for bringing this to my attention, so those lower bounds could be updated 15 years after the fact.

In 2014, Daniel Horsley wrote a paper Generalizing Fisher's Inequality to coverings and packings, which improved lower bounds for a wide range of coverings with t=2. In June 2017, he and Rakhi Singh wrote a paper New lower bounds for t-coverings generalizing the results to other values of t. Together, those results improved many hundreds of lower bound results in the tables.

In the process of incorporating the new results, I updated the way lower bounds are handled. In addition to the lower bound, there is now a note on how the lower bound was achieved. This is left blank for ones following from the Schonheim bound given below, but a brief reference for all other lower bounds is given.

Lower Bounds (older results)

More recently, in 2002 David Applegate, Eric Rains, and Neil Sloane showed that C(10,5,4)=51;  C(11,6,5)>=96;  C(11,7,6)=84;  C(12,7,6)>=165;  C(12,8,7)=126;  C(13,9,8)=185.

Also in 2002, John Bate, Ben Li, and John van Rees showed that C(19,6,2)=15, and Li and van Rees showed C(17,10,3)=11.

In 2004, Malcolm Greig, Li, and van Rees showed that C(28,9,2)=C(41,13,2)=14.

In 2007, Klas Marström gave many new lower bounds, and found all non-isomorphic coverings for many parameters.

These bounds, with other improvements that result from them, are given here.

Below are the improvements that we know of on lower bounds for the covering designs in the tables of our JCD paper New Constructions for Covering Designs, since that paper appeared in 1995.  These improvements appear roughly in the order we learned of them.  For each, we give the date (either when we learned of the result, or a publication date) and attribution, along with the old lower bound. We also give the improvements that follow directly from the Schonheim inequality:

      C(v,k,t)  >=  ceiling( (v/k) * C(v-1,k-1,t-1) )

And we state a lower bound as an equality if we know of an actual covering whose size matches the bound.