Achievable Rates and Algorithms for Group Testing with Runlength Constraints

TL;DR

This paper analyzes bounds on superimposed codes with runlength constraints for non-adaptive group testing, improving existing constructions and proposing randomized algorithms for code generation.

Contribution

It refines bounds on superimposed codes with runlength constraints, introduces improved constructions using the Lovász Local Lemma, and suggests randomized algorithms for code creation.

Findings

Improved bounds on code length with runlength constraints.

New constructions outperform previous methods.

Proposed $O(n^k)$ randomized algorithms for code construction.

Abstract

In this paper, we study bounds on the minimum length of $(k,n,d)$-superimposed codes introduced by Agarwal et al. [1], in the context of Non-Adaptive Group Testing algorithms with runlength constraints. A $(k,n,d)$-superimposed code of length $t$ is a $t \times n$ binary matrix such that any two 1's in each column are separated by a run of at least $d$ 0's, and such that for any column $\mathbf{c}$ and any other $k-1$ columns, there exists a row where $\mathbf{c}$ has $1$ and all the remaining $k-1$ columns have $0$. Agarwal et al. proved the existence of such codes with $t=\Theta(dk\log(n/k)+k^2\log(n/k))$. Here we investigate more in detail the coefficients in front of these two main terms as well as the role of lower order terms. We show that improvements can be obtained over the construction in [1] by using different constructions and by an appropriate exploitation of the Lov\'asz Local Lemma in this context. Our findings also suggest $O(n^k)$ randomized Las Vegas algorithms for the construction of such codes. We also extend our results to Two-Stage Group Testing algorithms with runlength constraints.