A Semi-Random Construction of Small Covering Arrays · arXivDesk
1703.05252 Mar 15, 2017 20 pages. This version subsumes the results in the previous version, "Families of Mass Destruction," generalising the results from set systems to covering arrays over larger alphabets
A Semi-Random Construction of Small Covering Arrays Shagnik Das, Tamás Mészáros
Abstract
Given a set S S S of v ≥ 2 v \ge 2 v ≥ 2 symbols, and integers k ≥ t ≥ 2 k \ge t \ge 2 k ≥ t ≥ 2 and N ≥ 1 N \ge 1 N ≥ 1
, an
array
A ∈ S N × k A \in S^{N \times k} A ∈ S N × k is an
( N ; t , k , v ) (N; t, k, v) ( N ; t , k , v ) -covering array if all sequences in
appear as rows in every
subarray of
. These arrays have a wide variety of applications, driving the search for small covering arrays. The covering array number,
C A N ( t , k , v ) \mathrm{CAN}(t,k,v) CAN ( t , k , v ) , is the smallest
for which an
( N ; t , k , v ) (N; t,k,v) ( N ; t , k , v ) -covering array exists. In this paper, we combine probabilistic and linear algebraic constructions to improve the upper bounds on
C A N ( t , k , v ) \mathrm{CAN}(t,k,v) CAN ( t , k , v ) by a factor of
, showing that for prime powers
,
C A N ( t , k , v ) ≤ ( 1 + o ( 1 ) ) ( ( t − 1 ) v t / ( 2 log 2 v − log 2 ( v + 1 ) ) ) log 2 k \mathrm{CAN}(t,k,v) \le (1 + o(1)) \left( (t-1) v^t / (2 \log_2 v - \log_2 (v+1)) \right)\log_2 k CAN ( t , k , v ) ≤ ( 1 + o ( 1 )) ( ( t − 1 ) v t / ( 2 log 2 v − log 2 ( v + 1 )) ) log 2 k , which also offers improvements for large
that are not prime powers. Our main tool, which may be of independent interest, is a construction of an array with
rows that covers the maximum possible number of subsets of size
.
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