Improved Probabilistic Lower Bounds for Separable Matrices

Abstract

This work focuses on non-adaptive combinatorial group testing, with a primary goal of efficiently identifying a set of at most $d$ defective elements among a given set of $n$ elements using the fewest possible tests. Non-adaptive combinatorial group testing often employs disjunctive matrices (DM) and separable matrices (SM). This paper discusses separable matrices and recently introduced list-decoding separable matrices (LDSM) with list size $n^{1/d}$, which allow for non-adaptive identification of defectives with the decoding complexity linear in the number of tests and the number of elements. In our study, we distinguish two subclasses of these matrices: matrices which can be used when the number of defectives $d$ is a priori known ($d$-SM and $(d, n^{1/d})$-LDSM), and matrices which can be used for any subset of at most $d$ defectives ($\bar{d}$-SM and $(\bar{d}, n^{1/d})$-LDSM). Our contribution lies in deriving new lower bounds on the rates of $d$-SM, $\bar{d}$-SM, $(d, n^{1/d})$-LDSM and $(\bar{d}, n^{1/d})$-LDSM for an arbitrary number $d \ge 3$ of defectives.