Journal of Combinatorial Theory Series A ( IF 0.9 ) Pub Date : 2023-11-28 , DOI: 10.1016/j.jcta.2023.105834
Dor Elimelech , Moshe Schwartz
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 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 ∞.
中文翻译:
二阶足球池问题与广义覆盖码最优率
经典足球彩池问题的目标是确定要购买多少张彩票,以保证至少从一系列n场比赛中正确猜测。我们研究这个问题的广义(二阶)版本,其中这n 个游戏中的任何一个都包含两个子游戏。足球池问题的二阶版本是使用广义覆盖半径的概念来表述的,最近提出了广义覆盖半径作为线性码的基本属性。我们考虑将此属性扩展到一般(不一定是线性)码,并通过在给定固定归一化覆盖半径的情况下找到二阶覆盖码的最优速率函数来为我们的问题提供渐近解决方案。我们还证明了当码长趋于无穷大时,二阶覆盖码在码率足够大的码中所占的比例趋于1。