Quantitative Frameproof Codes and Hypergraphs

TL;DR

This paper introduces and analyzes a quantitative extension of frameproof codes and hypergraphs, providing asymptotically optimal bounds and exact sizes for various parameters, and generalizing the Erdős matching number.

Contribution

It presents the first asymptotic bounds and exact size results for quantitative frameproof codes and hypergraphs, extending classical combinatorial concepts.

Findings

Derived asymptotically optimal bounds for maximum sizes.

Determined exact sizes for a broad range of parameters.

Introduced a generalized Erdős matching number and provided estimates.

Abstract

Frameproof codes are a class of secure codes introduced by Boneh and Shaw in the context of digital fingerprinting, and have been widely studied from a combinatorial point of view. In this paper, we study a quantitative extension of frameproof codes and hypergraphs, referred to as {\it quantitative frameproof codes and hypergraphs}. We give asymptotically optimal bounds on the maximum sizes of these structures and determine their exact sizes for a broad range of parameters. In particular, we introduce a generalized version of the Erd\H{o}s matching number in our proof and derive relevant estimates for it.