二阶足球球问题及广义覆盖码的最优率

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A Pub Date : 2023-11-28 DOI:10.1016/j.jcta.2023.105834

Dor Elimelech , Moshe Schwartz

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引用次数: 0

摘要

经典足球问题的目标是确定要买多少张彩票，以保证在进行的n场比赛中至少有n - r次正确猜测。我们研究了这个问题的广义(二阶)版本，其中这n个对策中的任何一个都由两个子对策组成。足球台球问题的二阶版本是用广义覆盖半径的概念来表述的，广义覆盖半径是最近提出的线性码的基本性质。我们考虑将这一性质推广到一般(不一定是线性)码，并在给定固定的归一化覆盖半径的情况下，通过寻找二阶覆盖码的最优速率函数，给出了问题的渐近解。我们还证明了当码长趋于∞时，在足够大码率的码中二阶覆盖码的分数趋于1。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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

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 ∞.

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来源期刊

Journal of Combinatorial Theory Series A 数学-数学

CiteScore

2.90

自引率

9.10%

发文量

审稿时长

12 months

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.