Near optimal constructions of frameproof codes
Miao Liu, Zengjiao Ma, Chong Shangguan
TL;DR
This work resolves the asymptotic size $M_{c,l}(q)$ of $q$-ary $(c,l)$-frameproof codes by linking the problem to the Erdős Matching Conjecture and Blackburn's limit $R_{c,l}$. It provides a probabilistic lower bound that matches Blackburn's upper bound up to leading order, establishing an exact expression for $R_{c,l}$ for all fixed $c,l$ and giving a refined upper bound for $M_{c,l}(q)$. The approach hinges on translating frameproof codes into $c$-cover-free, $l$-partite, $l$-uniform hypergraphs and applying near-optimal hypergraph packings via the Frankl–Rödl/Pippenger framework, including a new induced, faithful packing theorem. This bridges secure coding theory with extremal set theory, yielding a comprehensive solution to Blackburn’s open problem and advancing the methodology for non-explicit probabilistic constructions in combinatorial coding theory.
Abstract
Frameproof codes are a class of secure codes that were originally introduced in the pioneering work of Boneh and Shaw in the context of digital fingerprinting. They can be used to enhance the security and credibility of digital content. Let $M_{c,l}(q)$ denote the largest cardinality of a $q$-ary $c$-frameproof code with length $l$. Based on an intriguing observation that relates $M_{c,l}(q)$ to the renowned Erdős Matching Conjecture in extremal set theory, in 2003, Blackburn posed an open problem on the precise value of the limit $R_{c,l}=\lim_{q\rightarrow\infty}\frac{M_{c,l}(q)}{q^{\lceil l/c \rceil}}$. By combining several ideas from the probabilistic method, we present a lower bound for $M_{c,l}(q)$, which, together with an upper bound of Blackburn, completely determines $R_{c,l}$ for {\it all} fixed $c,l$, and resolves the above open problem in the full generality. We also present an improved upper bound for $M_{c,l}(q)$.