List-Decoding Separable Matrices for Non-Adaptive Combinatorial Group Testing

In this paper, we introduce a new concept of list-decoding separable matrices with strength
$d$
and list size
$L$
, denoted as
$(\overline{d}, L)$
-LDSM, for non-adaptive combinatorial group testing, which is a generalized notion of several types of known test matrices such as d-disjunct matrices,
$\overline{d}$
-separable matrices, and d-union-free codes with fast decoding. We first provide a two-step identifying algorithm for
$(\overline{d}, L)$
-LDSM of size
$n\times m$
and show that it can determine all the
$\leq d$
positives among
$m$
items in time
$O(\max\{nm,\ nL^{d}\})$
. Furthermore, we derive a lower bound on the largest rate of
$(\overline{d},\ L)$
-LDSM for any
$L\geq d\geq 2$
by random coding with expurgation. In particular, when
$L=m 1/d$
, the identifying algorithm of
$(d,\ m^{1/\overline{d}})$
-LDSM is as efficient as that for d-disjunct matrices, while the rate of
$(\overline{d},\ m^{1/d})$
-LDSM can be significantly larger than that of d-disjunct matrices.