二阶足球台球问题与广义覆盖码的最优覆盖率
The Second-Order Football-Pool Problem and the Optimal Rate of Generalized-Covering Codes
摘要：
经典的足球台球问题的目标是确定要购买多少张彩票，以保证至少$n-r$在一系列$n$游戏中猜对。我们研究了这个问题的一个广义（二阶）版本，其中任何这些$n$游戏由两个子游戏。足球池问题的二阶版本制定使用的概念，最近提出的线性码的一个基本属性的广义覆盖半径。我们认为这个属性的一般（不一定是线性）代码的扩展，并提供了一个渐近解决方案，我们的问题，找到最佳的二阶覆盖码的速率函数给定一个固定的归一化覆盖半径。我们还证明了当码长趋于$\infty$时，在足够大的码率的码中，二阶覆盖码的分数趋于$1$。
英文摘要：
The goal of the classic football-pool problem is to determine how many lottery tickets are to be bought in order to guarantee at least $n-r$ correct guesses out of a sequence of $n$ games played. We study a generalized (second-order) version of this problem, in which any of these $n$ games consists of two sub-games. The second-order version of the football-pool problem is formulated using the notion of generalized-covering radius, recently proposed as a fundamental property of linear codes. We consider an extension of this property to general (not necessarily linear) codes, and provide an asymptotic solution to our problem by finding the optimal rate function of second-order covering codes given a fixed normalized covering radius. We also prove that the fraction of second-order covering codes among codes of sufficiently large rate tends to $1$ as the code length tends to $\infty$.
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