良好的可追溯性代码确实存在
Good traceability codes do exist
摘要：
可追踪码是Chor、Fiat和Naor在1994年提出的一种组合对象，用于在叛逆者追踪方案中追踪数字内容的来源。令$F$是大小为$q$的字母集，并且$n$是正整数。$t$ -可追溯性代码是代码$\mathscr{C}\subseteq F^n$，其可用于从至多$t$叛徒的共谋中捕获至少一个共谋者。已证明$q\le t$不存在$t$ -可追溯性代码。当$q>t^2$、$t$ -具有正码率的可追溯码可以从具有大的最小距离的纠错码构造。因此，Barg和Kabatiansky在2004年询问是否存在$t+1\le q\le t^2$具有正码率的$t$ -可追溯性代码。2010年，Blackburn、Etzion和Ng使用概率方法对$q\ge t^2-\lceil t/2\rceil+1$的这个问题给出了肯定的答案。然而，他们没有看到如何使用他们的概率方法来回答$q$剩余值的这个问题。他们甚至怀疑，可能有一个“Plotkin界限”的可追溯性代码，禁止这种代码的存在。本文对Barg-Kabatiansky的问题给出了一个完整的回答（肯定的）。令人惊讶的是，我们的结构是确定性的。
英文摘要：
Traceability codes are combinatorial objects introduced by Chor, Fiat and Naor in 1994 to be used to trace the origin of digital content in traitor tracing schemes. Let $F$ be an alphabet set of size $q$ and $n$ be a positive integer. A $t$-traceability code is a code $\mathscr{C}\subseteq F^n$ which can be used to catch at least one colluder from a collusion of at most $t$ traitors. It has been shown that $t$-traceability codes do not exist for $q\le t$. When $q>t^2$, $t$-traceability codes with positive code rate can be constructed from error correcting codes with large minimum distance. Therefore, Barg and Kabatiansky asked in 2004 that whether there exist $t$-traceability codes with positive code rate for $t+1\le q\le t^2$. In 2010, Blackburn, Etzion and Ng gave an affirmative answer to this question for $q\ge t^2-\lceil t/2\rceil+1$, using the probabilistic methods. However, they did not see how their probabilistic methods can be used to answer this question for the remaining values of $q$. They even suspected that there may be a `Plotkin bound' of traceability codes that forbids the existence of such codes. In this paper, we give a complete answer to Barg-Kabatiansky's question (in the affirmative). Surprisingly, our construction is deterministic.
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。
