一般有界通道上的两批说谎者对策
Two-batch liar games on a general bounded channel
摘要：
我们考虑一个扩展的2人Rényi-Ulam说谎者游戏中的谎言是由一个通道$C$，一组允许的谎言字符串的最大长度$k$。Carole选择$x\in[n]$，Paul进行$t$ -ary查询以唯一地确定$x$。在每个$q$轮中，Paul弱划分$[n]=A_0\cup >... \cup A_{t-1}$并请求$a$，使得$x\in A_a$。Carole用一些$b$进行响应，如果是$a\neq b$，则$x$积累一个谎言$(a,b)$。Carole的$x$谎言字符串必须在通道$C$中。如果保罗在$q$回合内确定$x$，他就赢了。我们进一步限制Paul在两个离线批次中提出他的问题。我们表明，对于第二批的大小范围，搜索空间$[n]$的最大大小，保罗可以保证找到的区别元素是$\sim t^{q+k}/(E_k(C)\binom{q}{k})$作为$q\to\infty$，其中$E_k(C)$是谎言字符串的数量在$C$的最大长度$k$。这概括了Dumitriu和Spencer以及Ahlswede，Cicalese和Deppe以前的工作。我们扩展了Paul的策略，以统一的方式解决了病态说谎者变体，从而给出了通道$C$的渐进完美两批自适应码的存在性。
英文摘要：
We consider an extension of the 2-person Rényi-Ulam liar game in which lies are governed by a channel $C$, a set of allowable lie strings of maximum length $k$. Carole selects $x\in[n]$, and Paul makes $t$-ary queries to uniquely determine $x$. In each of $q$ rounds, Paul weakly partitions $[n]=A_0\cup >... \cup A_{t-1}$ and asks for $a$ such that $x\in A_a$. Carole responds with some $b$, and if $a\neq b$, then $x$ accumulates a lie $(a,b)$. Carole's string of lies for $x$ must be in the channel $C$. Paul wins if he determines $x$ within $q$ rounds. We further restrict Paul to ask his questions in two off-line batches. We show that for a range of sizes of the second batch, the maximum size of the search space $[n]$ for which Paul can guarantee finding the distinguished element is $\sim t^{q+k}/(E_k(C)\binom{q}{k})$ as $q\to\infty$, where $E_k(C)$ is the number of lie strings in $C$ of maximum length $k$. This generalizes previous work of Dumitriu and Spencer, and of Ahlswede, Cicalese, and Deppe. We extend Paul's strategy to solve also the pathological liar variant, in a unified manner which gives the existence of asymptotically perfect two-batch adaptive codes for the channel $C$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
