{"title":"The second-order football-pool problem and the optimal rate of generalized-covering codes","authors":"Dor Elimelech , Moshe Schwartz","doi":"10.1016/j.jcta.2023.105834","DOIUrl":null,"url":null,"abstract":"<div><p>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 <span><math><mi>n</mi><mo>−</mo><mi>r</mi></math></span> correct guesses out of a sequence of <em>n</em> games played. We study a generalized (second-order) version of this problem, in which any of these <em>n</em><span> 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 ∞.</span></p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"203 ","pages":"Article 105834"},"PeriodicalIF":0.9000,"publicationDate":"2023-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Combinatorial Theory Series A","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0097316523001024","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
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 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 ∞.
期刊介绍：
The Journal of Combinatorial Theory publishes original mathematical research concerned with theoretical and physical aspects of the study of finite and discrete structures in all branches of science. Series A is concerned primarily with structures, designs, and applications of combinatorics and is a valuable tool for mathematicians and computer scientists.
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号
