New Bounds for Frameproof Codes

Frameproof codes are used to fingerprint digital data. They can prevent copyrighted materials from unauthorized use. In this paper, we study upper and lower bounds for <inline-formula> <tex-math notation="LaTeX">$w$ </tex-math></inline-formula>-frameproof codes of length <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> over an alphabet of size <inline-formula> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula>. The upper bound is based on a combinatorial approach and the lower bound is based on a probabilistic construction. Both bounds can improve one of the previous results when <inline-formula> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> is small compared with <inline-formula> <tex-math notation="LaTeX">$w$ </tex-math></inline-formula>, say <inline-formula> <tex-math notation="LaTeX">$cq\leq w$ </tex-math></inline-formula> for some constant <inline-formula> <tex-math notation="LaTeX">$c\leq q$ </tex-math></inline-formula>. Furthermore, we pay special attention to binary frameproof codes. We show a binary <inline-formula> <tex-math notation="LaTeX">$w$ </tex-math></inline-formula>-frameproof code of length <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> cannot have more than <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> codewords if <inline-formula> <tex-math notation="LaTeX">$N<\binom {w+1}{2}$ </tex-math></inline-formula>.

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