Partial covering arrays for data hiding and quantization

TL;DR

This paper introduces partial covering arrays in the Boolean cube to optimize data hiding and quantization, providing bounds and solutions related to linear codes, with implications for cryptography and data compression.

Contribution

It formulates the partial covering array problem, derives asymptotic bounds, and finds solutions within linear codes, linking the problem to cryptography and quantization efficiency.

Findings

The ratio of k-faces intersected by the set S approaches a bound less than 1 as n increases.

A solution within the class of linear codes is provided.

Connections to cryptography and quantization efficiency are discussed.

Abstract

We consider the problem of finding a set (partial covering array) $S$ of vertices of the Boolean $n$-cube having cardinality $2^{n-k}$ and intersecting with maximum number of $k$-dimensional faces. We prove that the ratio between the numbers of the $k$-faces containing elements of $S$ to $k$-faces is less than $1-\frac{1+o(1)}{\sqrt{2\pi k}}$ as $n\rightarrow\infty$ for sufficiently large $k$. The solution of the problem in the class of linear codes is found. Connections between this problem, cryptography and an efficiency of quantization are discussed.