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

TL;DR

This paper extends the classic football-pool problem to a second-order version involving sub-games, using generalized-covering radius to analyze the optimal rate of codes, and shows most codes are optimal at large lengths.

Contribution

It introduces a second-order football-pool problem framework using generalized-covering radius and determines the asymptotic optimal rate function for such codes.

Findings

Derived the asymptotic optimal rate function for second-order covering codes.

Proved that the fraction of optimal second-order covering codes approaches 1 as code length increases.

Extended the notion of covering radius to non-linear codes in the context of the problem.

Abstract

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 $\infty$.