Various kinds of fingerprinting codes and their related combinatorial structures are extensively studied for protecting copyrighted materials. This paper concentrates on one specialised fingerprinting code named wide-sense frameproof codes in order to prevent innocent users from being framed. Let $Q$ be a finite alphabet of size $q$. Given a $t$-subset $X=\{x ^1,\ldots, x ^t\}\subseteq Q^n$, a position $i$ is called undetectable for $X$ if the values of the words of $X$ match in their $i$th position: $x_i^1=\cdots=x_i^t$. The wide-sense descendant set of $X$ is defined by $\wdesc(X)=\{y\in Q^n:y_i=x_i^1,i\in {U}(X)\},$ where ${U}(X)$ is the set of undetectable positions for $X$. A code ${\cal C}\subseteq Q^n$ is called a wide-sense $t$-frameproof code if $\wdesc(X) \cap{\cal C} = X$ for all $X \subseteq {\cal C}$ with $|X| \le t$. The paper improves the upper bounds on the sizes of wide-sense $2$-frameproof codes by applying techniques on non $2$-covering Sperner families and intersecting families in extremal set theory.

中文翻译：

---

广义 2-frameproof 代码

各种指纹代码及其相关的组合结构被广泛研究以保护受版权保护的材料。为了防止无辜用户被陷害，本文集中研究一种称为广义防框码的专用指纹识别码。令 $Q$ 是大小为 $q$ 的有限字母表。给定一个 $t$-subset $X=\{x ^1,\ldots, x ^t\}\subseteq Q^n$，如果$X$ 匹配第 $i$ 个位置：$x_i^1=\cdots=x_i^t$。$X$ 的广义后代集定义为 $\wdesc(X)=\{y\in Q^n:y_i=x_i^1,i\in {U}(X)\},$ 其中 $ {U}(X)$ 是 $X$ 的不可检测位置的集合。代码 ${\cal C}\subseteq Q^n$ 被称为广义 $t$-frameproof 代码，如果 $\wdesc(X) \cap{\cal C} = X$ for all $X \subseteq { \cal C}$ 与 $|X| \le t$。