The second-order football-pool problem and the optimal rate of generalized-covering codes

Dor Elimelech, Moshe Schwartz

Research output: Contribution to journalArticlepeer-review

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 ∞.

Original languageEnglish
Article number105834
JournalJournal of Combinatorial Theory. Series A
Volume203
DOIs
StatePublished - 1 Apr 2024

Keywords

  • Covering codes
  • Football-pool problem
  • Generalized covering radius

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics

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