Improved Lower Bounds for Secure Codes and Related Structures

TL;DR

This paper advances the understanding of secure codes by establishing improved lower bounds for their rates and related structures, especially for larger alphabet sizes and higher parameters, enhancing their theoretical limits.

Contribution

It introduces new lower bounds for secure codes and related combinatorial structures, and presents a general method for deriving bounds for higher parameters.

Findings

Improved lower bounds for 2-frameproof codes.

Enhanced bounds for strongly 2-separable matrices.

A general method for bounds when t ≥ 3.

Abstract

Secure codes are widely-studied combinatorial structures which were introduced for traitor tracing in broadcast encryption. To determine the maximum size of such structures is the main research objective. In this paper, we investigate the lower bounds for secure codes and their related structures. First, we give some improved lower bounds for the rates of $2$-frameproof codes and $\overline{2}$-separable codes for slightly large alphabet size. Then we improve the lower bounds for the rate of some related structures, i.e., strongly $2$-separable matrices and $2$-cancellative set families. Finally, we give a general method to derive new lower bounds for strongly $t$-separable matrices and $t$-cancellative set families for $t\ge 3.$