Improved upper bounds for wide-sense frameproof codes

TL;DR

The paper addresses upper bounds for wide-sense $t$-frameproof codes, bridging digital fingerprinting with extremal set theory. It combines Sperner theory, shadows, and local LYM tools to strengthen the $t=2$ bound, and establishes a general upper bound for $t\ge3$ by connecting codes to $(t-1)$-cover-free families; a tight bound is achieved for small lengths, with a complete characterization of optimal codes as permutation matrices in standard form. These results not only tighten several existing bounds, especially in the binary case, but also illuminate structural properties of optimal wide-sense frameproof codes. The methods advance the interplay between coding theory and combinatorial set systems, with potential implications for noise-tolerant fingerprinting and related combinatorial design problems.

Abstract

Frameproof codes have been extensively studied for many years due to their application in copyright protection and their connection to extremal set theory. In this paper, we investigate upper bounds on the cardinality of wide-sense $t$-frameproof codes. For $t=2$, we apply results from Sperner theory to give a better upper bound, which significantly improves a recent bound by Zhou and Zhou. For $t\geq 3$, we provide a general upper bound by establishing a relation between wide-sense frameproof codes and cover-free families. Finally, when the code length $n$ is at most $\frac{15+\sqrt{33}}{24}(t-1)^2$, we show that a wide-sense $t$-frameproof code has at most $n$ codewords, and the unique optimal code consists of all weight-one codewords. As byproducts, our results improve several best known results on binary $t$-frameproof codes.

Theorems & Definitions (60)

• Definition 1
• Theorem 1: Zhou-Zhou
• Theorem 2
• Corollary 1
• Theorem 3: Stinson-Wei1998
• Theorem 4: Blackburn2003
• Theorem 5: Shangguan2017
• Theorem 6
• Corollary 2
• Theorem 7