Improved Constructions of Frameproof Codes

TL;DR

This paper introduces a recursive method to construct $c$-frameproof codes with improved parameters, establishing existence results and exact asymptotic ratios for certain code lengths and alphabet sizes.

Contribution

It provides a new recursive construction for $c$-frameproof codes and determines their asymptotic ratios for specific lengths and alphabet sizes, advancing coding theory.

Findings

Established existence of $q$-ary $c$-frameproof codes of length $c+2$ with specific sizes.

Proved that $R_{c,c+2}=(c+2)/c$ for certain prime power conditions.

Achieved upper bound matching ratios for codes when $c+1$ is a prime power.

Abstract

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 $R_{c,l}=\lim_{q\rightarrow \infty}M_{c,l}(q)/q^{\lceil l/c\rceil}$. It has been determined by Blackburn that $R_{c,l}=1$ when $l\equiv 1\ (\bmod\ c)$, $R_{c,l}=2$ when $c=2$ and $l$ is even, and $R_{3,5}=5/3$. 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 $\frac{c+2}{c}(q-1)^2+1$ for all odd $q$ when $c=2$ and for all $q\equiv 4\pmod{6}$ 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.