Mixed covering arrays of strength three with few factors

Covering arrays with mixed alphabet sizes, or mixed covering arrays, are useful generalizations of covering arrays that are motivated by software and network testing. Suppose that there are k factors, and that the ith factor takes values or levels from a set Gi of size gi. A run is an assignment of an admissible level to each factor. A mixed covering array, MCA(N;t,k,g1g2…gk)(N;t,k,g1g2…gk), is a collection of N runs such that for any t   distinct factors, i1,i2,…,iti1,i2,…,it, every t  -tuple from Gi1×Gi2×⋯×GitGi1×Gi2×⋯×Git occurs in factors i1,i2,…,iti1,i2,…,it in at least one of the N   runs. When g=g1=g2=⋯=gkg=g1=g2=⋯=gk, an MCA(N;t,k,g1g2…gk)(N;t,k,g1g2…gk) is a CA(N;t,k,g)(N;t,k,g). The mixed covering array number, denoted by MCAN(t,k,g1g2…gk)(t,k,g1g2…gk), is the minimum N for which an MCA(N;t,k,g1g2…gk)(N;t,k,g1g2…gk) exists. In this paper, we focus on the constructions of mixed covering arrays of strength three. The numbers MCAN(3,k,g1g2…gk)(3,k,g1g2…gk) are determined for all cases with k∈{3,4}k∈{3,4} and for most cases with k∈{5,6}k∈{5,6}.