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A covering array [Formula: see text] is an [Formula: see text] array such that every [Formula: see text] subarray covers at least once each [Formula: see text]-tuple from [Formula: see text] symbols. For given [Formula: see text], [Formula: see text], and [Formula: see text], the minimum number of rows for which exists a CA is denoted by [Formula: see text] (CAN stands for Covering Array Number) and the corresponding CA is optimal. Optimal covering arrays have been determined algebraically for a small subset of cases; but another alternative to find CANs is the use of computational search. The present work introduces a new orderly algorithm to construct non-isomorphic covering arrays; this algorithm is an improvement of a previously reported algorithm for the same purpose. The construction of non-isomorphic covering arrays is used to prove the nonexistence of certain covering arrays whose nonexistence implies the optimality of other covering arrays. From the computational results obtained, the following CANs were established: [Formula: see text] for [Formula: see text], [Formula: see text], and [Formula: see text]. In addition, the new result [Formula: see text], and the already known existence of [Formula: see text], imply [Formula: see text].

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Covering perfect hash families (CPHFs) are combinatorial designs that represent certain covering arrays in a compact way. In previous works, CPHFs have been constructed using backtracking, tabu search, and greedy algorithms. Backtracking is convenient for small CPHFs, greedy algorithms are appropriate for large CPHFs, and metaheuristic algorithms provide a balance between execution time and quality of solution for small and medium-size CPHFs. This work explores the construction of CPHFs by means of a simulated annealing algorithm. The neighborhood function of this algorithm is composed of three perturbation operators which together provide exploration and exploitation capabilities to the algorithm. As main computational results we have the generation of 64 CPHFs whose derived covering arrays improve the best-known ones. In addition, we use the simulated annealing algorithm to construct quasi-CPHFs from which quasi covering arrays are derived that are then completed and postoptimized; in this case the number of new covering arrays is 183. Together, the 247 new covering arrays improved the upper bound of 683 covering array numbers.

A covering array CA([Formula: see text]) of strength [Formula: see text] and order [Formula: see text] is an [Formula: see text] array over [Formula: see text] with the property that every [Formula: see text] subarray covers all members of [Formula: see text] at least once. When the value of [Formula: see text] is the minimum possible it is named as the covering array number (CAN) i.e. [Formula: see text]. Two CAs are isomorphic if one of them can be derived from the other by a combination of a row permutation, a column permutation, and a symbol permutation in a subset of columns. Isomorphic CAs have equivalent coverage properties, and can be considered as the same CA; the truly distinct CAs are those which are non-isomorphic among them. An interesting and hard problem is to construct all the non-isomorphic CAs that exist for a particular combination of the parameters [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]. We constructed the non-isomorphic CAs for 70 combinations of values of the parameters [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text], the results allow us to determine CAN(3,13,2) =16, CAN(3,14,2) =16, CAN(3,15,2) =17, CAN(3,16,2) =17, and CAN(2,10,3) =14. The exact lower bound for these covering arrays numbers had not been determined before either by computational search or by algebraic analysis.

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