Local Search and a Construction Scan for Covering-Design and Lottery-Covering Tables

A covering design \(C(v,k,t)\) is a family of \(k\)-subsets (blocks) of a \(v\)-set containing every \(t\)-subset; a lottery covering \(C(v,k,t,m)\) only requires every \(m\)-subset to meet some block in at least \(t\) points. We describe the methods behind about a thousand improvements of the public tables of both. For covering designs, a decremental tabu search with clause weighting, as in set-cover local search, keeps exact move scores incrementally. For lottery coverings, the same search scores moves at the level of \(t\)-subsets by aggregating \(m\)-subset weights, so that a move is scored with as many lookups as for covering designs (updating after a move remains costly); we also describe engineering for covers with up to \(2\cdot 10^9\) \(m\)-subsets. Finally, a dynamic-programming scan applies standard recursive constructions to the whole live table, and the explicit products are then reduced by the search. We evaluated the searches in a test whose protocol and sample were fixed in advance: 300 random covering-design entries and 300 random lottery entries not held by us, two minutes per run. The weighted search improved 14% of the covering-design entries and the same search without weighting 8%. On lottery entries the methods improved 1.3–4.3%, and the \(t\)-level search did better without weighting than with it. On 25 October 2026 the Covering Repository held 998 of our coverings: 401 covering designs and 65 lottery coverings found by search, and 532 constructed coverings.