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The goal of the classic football-pool problem is to determine how manylottery tickets are to be bought in order to guarantee at least $n-r$ correctguesses out of a sequence of $n$ games played. We study a generalized(second-order) version of this problem, in which any of these $n$ gamesconsists of two sub-games. The second-order version of the football-poolproblem is formulated using the notion of generalized-covering radius, recentlyproposed as a fundamental property of linear codes. We consider an extension ofthis property to general (not necessarily linear) codes, and provide anasymptotic solution to our problem by finding the optimal rate function ofsecond-order covering codes given a fixed normalized covering radius. We alsoprove that the fraction of second-order covering codes among codes ofsufficiently large rate tends to $1$ as the code length tends to $\infty$.

The Second-Order Football-Pool Problem and the Optimal Rate of Generalized-Covering Codes