Almost Cover-Free Codes and Designs

TL;DR

This paper introduces the concept of almost cover-free codes, a probabilistic generalization relevant to combinatorial group testing, and provides lower bounds on their capacity using a random coding approach.

Contribution

It defines almost cover-free codes, connects them to group testing problems, and develops a random coding method to establish capacity bounds.

Findings

Introduces the concept of almost cover-free $(s,ll)$-codes.

Establishes lower bounds on the capacity of these codes.

Links the concept to nonadaptive group testing for defective sets.

Abstract

An $s$-subset of codewords of a binary code $X$ is said to be an {\em $(s,\ell)$-bad} in $X$ if the code $X$ contains a subset of other $\ell$ codewords such that the conjunction of the $\ell$ codewords is covered by the disjunctive sum of the $s$ codewords. Otherwise, the $s$-subset of codewords of $X$ is said to be an {\em $(s,\ell)$-good} in~$X$.mA binary code $X$ is said to be a cover-free $(s,\ell)$-code if the code $X$ does not contain $(s,\ell)$-bad subsets. In this paper, we introduce a natural {\em probabilistic} generalization of cover-free $(s,\ell)$-codes, namely: a binary code is said to be an almost cover-free $(s,\ell)$-code if {\em almost all} $s$-subsets of its codewords are $(s,\ell)$-good. We discuss the concept of almost cover-free $(s,\ell)$-codes arising in combinatorial group testing problems connected with the nonadaptive search of defective supersets (complexes). We develop a random coding method based on the ensemble of binary constant weight codes to obtain lower bounds on the capacity of such codes.