TY - JOUR

T1 - New bounds on the number of tests for disjunct matrices

AU - Shangguan, Chong

AU - Ge, Gennian

N1 - Publisher Copyright:
© 1963-2012 IEEE.

PY - 2016/12

Y1 - 2016/12

N2 - Given n items with at most d of which being positive, instead of testing these items individually, the theory of combinatorial group testing aims to identify all positive items using as few tests as possible. This paper is devoted to a fundamental and thirty-year-old problem in the nonadaptive group testing theory. A binary matrix is called d -disjunct if the Boolean sum of arbitrary d columns does not contain another column not in this collection. Let T(d) denote the minimal t , such that there exists a t × n,d -disjunct matrix with n>t. T(d) can also be viewed as the minimal t such that there exists a nonadaptive group testing scheme, which is better than the trivial one that tests each item individually. It was known that T(d)≥ d+22 and was conjectured that T(d)≥ (d+1)2. In this paper, we narrow the gap by proving T(d)/d2≥ (15+ 33)/24 , a quantity in [6/7,7/8].

AB - Given n items with at most d of which being positive, instead of testing these items individually, the theory of combinatorial group testing aims to identify all positive items using as few tests as possible. This paper is devoted to a fundamental and thirty-year-old problem in the nonadaptive group testing theory. A binary matrix is called d -disjunct if the Boolean sum of arbitrary d columns does not contain another column not in this collection. Let T(d) denote the minimal t , such that there exists a t × n,d -disjunct matrix with n>t. T(d) can also be viewed as the minimal t such that there exists a nonadaptive group testing scheme, which is better than the trivial one that tests each item individually. It was known that T(d)≥ d+22 and was conjectured that T(d)≥ (d+1)2. In this paper, we narrow the gap by proving T(d)/d2≥ (15+ 33)/24 , a quantity in [6/7,7/8].

KW - Nonadaptive group testing

KW - disjunct matrix

KW - graph matching number

UR - http://www.scopus.com/inward/record.url?scp=84999287155&partnerID=8YFLogxK

U2 - 10.1109/TIT.2016.2614726

DO - 10.1109/TIT.2016.2614726

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AN - SCOPUS:84999287155

SN - 0018-9448

VL - 62

SP - 7518

EP - 7521

JO - IEEE Transactions on Information Theory

JF - IEEE Transactions on Information Theory

IS - 12

M1 - 7580611

ER -