Improved Constructions of Frameproof Codes

摘要

Frameproof codes are used to preserve the security in the context of coalition when fingerprinting digital data. Let $M_{c,l}(q)$ be the largest cardinality of a $q$-ary $c$-frameproof code of length $l$ and $R_{c,l}=lim_{qrightarrow infty}M_{c,l}(q)/q^{lceil l/crceil}$. It has been determined by Blackburn that $R_{c,l}=1$ when $lequiv 1 (bmod c),$ $R_{c,l}=2$ when $c=2$ and $l$ is even, and $R_{3,5}={5over 3}$. In this paper, we give a recursive construction for $c$-frameproof codes of length $l$ with respect to the alphabet size $q$ . As applications of this construction, we establish the existence results for $q$-ary $c$-frameproof codes of length $c+2$ and size ${c+2over c}(q-1)^2+1$ for all odd $q$ when $c=2$ and for all $qequiv 4pmod{6}$ when $c=3$ . Furthermore, we show that $R_{c,c+2}=(c+2)/c$ meeting the upper bound given by Blackburn, for all integers $c$ such that $c+1$ is a prime power.