Improved bounds on 2-frameproof codes with length 4

Abstract

Frameproof codes (FPCs) are widely studied as they are classic fingerprinting codes that can protect copyrighted materials. The main interests are construction methods and bounds of the number of codewords of FPCs for a fixed length when the alphabet size approaches infinity. In this paper, we focus on the upper bound of the size of FPCs when the fixed length is 4 and the strength is 2. We obtain an upper bound $$2q^2-2q+7$$ on the size of a q-ary 2-FPC of length 4 for any positive integer $$q> 48$$ . The best previously well known bound of this type of FPCs is $$2q^2-2$$ , which is due to Blackburn (SIAM J Discret Math 16:499–510, 2003). Our new upper bound improves the previous upper bound and it is not very far from the current best lower bound $$2q^2-4q+3$$ obtained from the explicit construction due to Chee and Zhang (IEEE Trans Inf Theory 58:5449–5453, 2012).