On tight bounds for binary frameproof codes

TL;DR

This paper establishes tight bounds for binary $w$-frameproof codes, showing limitations on their size and structure for certain parameters, which advances understanding of their combinatorial properties.

Contribution

The paper provides new bounds and structural characterizations for binary $w$-frameproof codes, especially for the case when $w eq 2$, filling gaps in existing combinatorial code theory.

Findings

For all $w eq 2$, $SHF(N; n, 2, \\{1,w\\})$ exists only if $n \\leq N$.

When $n = N$, the $SHF$ must be a permutation matrix.

The bounds are tight for the specified parameter ranges.

Abstract

In this paper, we study $w$-frameproof codes, which are equivalent to $\{1,w\}$-separating hash families. Our main results concern binary codes, which are defined over an alphabet of two symbols. For all $w \geq 3$, and for $w+1 \leq N \leq 3w$, we show that an $SHF(N; n,2, \{1,w \})$ exists only if $n \leq N$, and an $SHF(N; N,2, \{1,w \})$ must be a permutation matrix of degree $N$.