The equivalence between optimal detecting arrays and super-simple OAs

Abstract

The notion of a detecting array (DTA) was proposed, recently, by Colbourn and McClary in their research on software interaction tests. Roughly speaking, testing with a (d, t)-DTA (N , k, v) can locate d interaction faults and detect whether there are more than d interaction faults. In this paper, we establish a general lower bound on sizes of DTAs and explore an equivalence between optimal DTAs and super-simple orthogonal arrays (OAs). Taking advantage of this equivalence, a great number of DTAs are constructed, which are all optimal in the sense of their sizes. In particular, an optimal (2, t)-DTA (N , 5, v) of strength t = 2 or 3 is shown to exist whenever v ≥ 3 excepting (t, v) ∈ {(2, 3), (2, 6), (3, 4), (3, 6)}.

Optimum mixed level detecting arrays

“…Combining Lemma 3.8 with Theorem 3.10 gives us the following useful corollary, which was first stated in Shi, Tang and Yin (2012).…”

“…Proof of Corollary 5.2. From Shi, Tang and Yin (2012), we know that an OA(t+1, k +1, m) implies the existence of a super-simple OA λ (t, k, m) with 2 ≤ λ ≤ m. So the conclusion (1) holds by taking u 1 = u 2 = · · · = u k = m in Theorem 5.1. The conclusion (2) follows directly from Theorem 5.1 by taking u 1 = u 2 = · · · = u k = m and d 2 = 0.…”

“…In addition, the jth level-set of cardinality v j is understood to be Z v j for 1 ≤ j ≤ k, unless otherwise stated. Under these conventions, we have the following lemma which is an analogue of Theorem 2.3 in Shi, Tang and Yin (2012). Hence, we omit its proof for brevity.…”

“…Taking v 1 = v 2 = · · · = v k = v in Theorem 3.2, we retrieve the lower bound on the function (d, t)-DTAN(k, v) given in Shi, Tang and Yin (2012).…”

“…It was proved in Shi, Tang and Yin (2012) that an optimum (d, t)-DTAN(N ; k, v) meeting the lower bound in (3.2) is equivalent to a supersimple OA d+1 (t, k, v). However, the structure of an optimum, mixed level DTA is much more complicated than that of an optimum, fixed level DTA even though the lower bounds in (3.1) and (3.2) look similar.…”