Wide-sense 2-frameproof codes

Abstract

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 ith position: $$x_i^1=\cdots =x_i^t$$ . The wide-sense descendant set of X is defined by $${{\text {wdesc}}}(X)=\{y\in Q^n:y_i=x_i^1,i\in {U}(X)\},$$ where U(X) is the set of undetectable positions for X. A code $$\mathcal{C}\subseteq Q^n$$ is called a wide-sense t-frameproof code if $${{\text {wdesc}}}(X) \cap \mathcal{C} = X$$ for all $$X \subseteq \mathcal{C}$$ 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.