Consider a lottery scheme consisting of randomly selecting a winning \(t\)-set from a universal \(m\)-set, while a player participates in the scheme by purchasing a playing set of any number of \(n\)-sets from the universal set prior to the draw, and is awarded a prize if \(k\) or more elements of the winning \(t\)-set occur in at least one of the player’s \(n\)-sets (\(1 \leq k \leq \{n,t\} \leq m\)). This is called a \(k\)-prize. The player may wish to construct a playing set, called a lottery set, which guarantees the player a \(k\)-prize, no matter which winning \(t\)-set is chosen from the universal set. The cardinality of a smallest lottery set is called the lottery number, denoted by \(L(m,n,t;k)\), and the number of such non-isomorphic sets is called the lottery characterisation number, denoted by \(\eta(m,n,t;k)\). In this paper, an exhaustive search technique is employed to characterise minimal lottery sets of cardinality not exceeding six, within the ranges \(2 \leq k \leq 4\), \(k \leq t \leq 11\), \(k \leq n \leq 12\), and \(\max\{n,t\} \leq m \leq 20\). In the process, \(32\) new lottery numbers are found, and bounds on a further \(31\) lottery numbers are improved. We also provide a theorem that characterises when a minimal lottery set has cardinality two or three. Values for the lottery characterisation number are also derived theoretically for minimal lottery sets of cardinality not exceeding three, as well as a number of growth and decomposition properties for larger lotteries.
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