$t$-Covering Arrays Generated by a Tiling Probability Model

Extensive surveys of covering arrays may be found in the papers of Sloane [5] and Colbourn [3]. Given t, m and the alphabet size |A|, we wish to find the minimum number of columns, n, such that there exists an m × n matrix that is t-covering. We will define N = N(m, t, α) as the smallest positive integer n such that there exists a covering array of dimensions m × n. At the Coimbra Zero-One Matrix Conference, the second author talked about the need to introduce new probability models to improve upper bounds on N(m, t, α) and the corresponding numbers for partial covering arrays [2]. In this paper we propose a specific way of doing so, once again using the Lovász local lemma as an auxiliary tool.

The Lovász Local Lemma ( [1]): Let C 1 , C 2 , . . . , C K be the events in arbitrary probability space. Suppose that each event C i is mutually independent of a set of all the other events C k but at most d, and that

Let R be the index set of all sets of t rows; |R| = m t . For r ∈ R, let C r be the event that the r th row set does not contain some vector v 1 , v 2 , . . . , v t in any of its columns. We wish to prove that P ( r∈R C ′ r ) > 0 if n ≥ N 0 , proving that N(m, t, α) ≤ N 0 . Now in [4] a general upper bound was provided on the size of covering arrays; this was

The proof used an elementary probability model that consisted of placing one letter of the alphabet independently in each of the mn positions with probability 1 α , i.e. by letting P (r i,j = x) = 1 α ∀x ∈ A. In the same paper, a special probability model was used, but only for the case α = 2, t = 3. Here the authors of [4], following the approach used in the doctoral thesis of Roux (see, e.g. [5]), used a probability model that independently places an equal number of zeros and ones in the rows of the matrix (the so-called “fixed weight rows” model.) Unfortunately this method becomes quite intractable in general, and it is our intent in this paper to explore a probability model that is, in some sense, intermediate between the general technique in [4] and the special method used there for α = 2, t = 3: Specifically, we seek to improve the general bound (1) using the method of placing consecutive and equally weighted tiles along the rows. We use tiles of dimension 1 × kα, such that there are exactly k x’s in each tile for each x ∈ A. By way of comparison, the general method used 1 × 1 tiles that led to a loss of control over the numbers of letters of each type in any row, while Roux’s method used a single long tile in each row, i.e., corresponded to k = n/2 (n even).

We consider two cases, when (i) k = 1 which yields an elementary equation relating N(m, t, α) and the variables m, t and α, and when (ii) k > 1, which yields better bounds as k increases, but which generates increasingly more complicated solutions. (i) We start with the case k = 1, and fill in our matrix using tiles that contain one randomly placed copy of each letter of the alphabet, assuming that α|n. Note that there are a total of α t possible vectors, and by the symmetry of our construction, all are equally likely to occur in selected rows. Thus P (C r ) ≤ λα t where λ is the probability that a specific vector z * = z 1 , z 2 , . . . , z t is missing in the set r of selected rows. Select an arbitrary set of t rows in the matrix. Consider the columns in any vertically aligned set of tiles. For each z i , there is exactly one value in any tile equal to z i , and α places it can be; moreover z * cannot occur in more than one column of the vertically stacked tiles in the selected rows. Therefore, the probability that z * is somewhere in these tiles is α • 1 α t = 1 α t-1 . Since there are n α tiles in any row of the m × n matrix, and the composition of these is determined independently, we have

and thus,

We can improve this bound slightly by using a technique found in [2], where the vectors z i = i, i, . . . , i ; 1 ≤ i ≤ α can be achieved for all sets r by including columns consisting of all i’s. There are α of these vectors; thus this reduces the number of z * ’s from α t to α tα. We can ignore these vectors in our calculation of P (C r ) so long as we remember to add α columns to our value N(m, t, α). So (2) may be improved as follows:

Our next step is to calculate d. For any set r of rows, there will be a dependency only on sets r 0 ∈ R such that r ∩ r 0 = ∅. We will bound the number of such r 0 ’s by choosing one row from r, and then choosing an arbitrary t-1 rows from the m-1 other rows in the matrix. Thus d ≤ t m-1 t-1 , so d + 1 ≤ tm t-1 (t-1)! . Substituting this i