Lower bounds for the football pool problem for 7 and 8 matches (Q872721)

Summary: Let \(k_3(n)\) denote the minimal cardinality of a ternary code of length \(n\) and covering radius one. In this paper we show \(k_3(7)\geq 156\) and \(k_3(8)\geq 402\) improving on the best previously known bounds \(k_3(7)\geq 153\) and \(k_3(8)\geq 398\). The proofs are founded on a recent technique of the author for dealing with systems of linear inequalities satisfied by the number of elements of a covering code, that lie in \(k\)-dimensional subspaces of \(\mathbb{F}_3^n\).