Frameproof codes have been extensively studied for many years due to their application in copyright protection and their connection to extremal set theory. In this paper, we investigate upper bounds on the cardinality of wide-sense $t$-frameproof codes. For $t=2$, we apply results from Sperner theory to give a better upper bound, which significantly improves a recent bound by Zhou and Zhou. For $t\geq 3$, we provide a general upper bound by establishing a relation between wide-sense frameproof codes and cover-free families. Finally, when the code length $n$ is at most $\frac{15+\sqrt{33}}{24}(t-1)^2$, we show that a wide-sense $t$-frameproof code has at most $n$ codewords, and the unique optimal code consists of all weight-one codewords. As byproducts, our results improve several best known results on binary $t$-frameproof codes.

翻译：框架防伪码因其在版权保护中的应用以及与极值集合论的联系，多年来被广泛研究。本文研究了广义$t$-框架防伪码基数（cardinality）的上界。对于$t=2$，我们应用Sperner理论给出了一个更优的上界，显著改进了Zhou与Zhou近期提出的界。对于$t\geq 3$，我们通过建立广义框架防伪码与无覆盖族（cover-free families）之间的关系，给出了一个通用上界。最后，当码长$n$不超过$\frac{15+\sqrt{33}}{24}(t-1)^2$时，我们证明广义$t$-框架防伪码至多包含$n$个码字，且唯一的最优码为全体重量为1的码字。作为副产品，我们的结果改进了二进制$t$-框架防伪码的若干已知最佳结论。