The minimum number of rows in covering arrays (equivalently, surjective codes) and radius-covering arrays (equivalently, surjective codes with a radius) has been determined precisely only in special cases. In this paper, explicit constructions for numerous best known covering arrays (upper bounds) are found by a combination of combinatorial and computational methods. For radius-covering arrays, explicit constructions from covering codes are developed. Lower bounds are improved upon using connections to orthogonal arrays, partition matrices, and covering codes, and in specific cases by computation. Consequently for some parameter sets the minimum size of a covering array is determined precisely. For certain of these, a complete classification of all inequivalent covering arrays is determined, again using computational techniques. Existence tables for up to 10 columns, up to 8 symbols, and all possible strengths are presented to report the best current lower and upper bounds, and classifications of inequivalent arrays.

Based on the classification of superregular matrices, the numbers of non-equivalent n-arcs and complete n-arcs in PG(r, q) are determined (i) for 4 q 19, 2 r q À 2 and arbitrary n, (ii) for 23 q 32, r ¼ 2 and n ! q À 8. The equivalence classes over both PGL (k, q) and PÀ ÀL(k, q) are considered throughout the examinations and computations. For the classification, an n-arc is represented by the systematic generator matrix of the corresponding MDS code, without the identity matrix part of it. A rectangular matrix like this is superregular, i.e., it has only non-singular square submatrices. Four types of superregular matrices are studied and the non-equivalent superregular matrices of different types are stored in databases. Some particular results on t(r, q) and m 0 (r, q)-the smallest and the second largest size for complete arcs in PG(r, q)-are also reported, stating that m 0 (2, 31) ¼ 22, m 0 (2, 32) ¼ 24, t(3, 23) ¼ 10, and m 0 (3, 23) ¼ 16. #

The minimum number of codewords in a code with t ternary and b binary coordinates and covering radius R is denoted by K(t,b,R). In this paper, necessary and sufficient conditions for K(t,b,R)=M are given for all M = 5. By the help of generalized s-surjective codes, we develop new methods for finding bounds for K(t,b,R). These results are used to prove the equality K(9,0,5)=6 as well as some new lower bounds such as K(2,7,3) =7,  K(3,6,3)=8,  K(5,3,3)=8, and K(9,0,4)=9. Some bounds for (nonmixed) quaternary codes are also obtained.

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The minimum number of codewords in a code with t ternary and b binary coordinates and covering radius R is denoted by K(t, b, R). In the paper, necessary and sufficient conditions for K(t, b, R) = M are given for M = 6 and 7 by proving that there exist exactly three families of optimal codes with six codewords and two families of optimal codes with seven codewords. The cases M 5 were settled in an earlier study by the same authors. For binary codes, it is proved that K(0, 2b + 4, b) 9 for b 1. For ternary codes, it is shown that K(3t + 2, 0, 2t) = 9 for t 2. New upper bounds obtained include K(3t + 4, 0, 2t) 36 for t 2. Thus, we have K(13, 0, 6) 36 (instead of 45, the previous best known upper bound).

The covering radius of binary 2-surjective codes of maximum length is studied in the paper. It is shown that any binary 2-surjective code of M codewords and of length n = M−1 (M−2)/2 has covering radius n 2 − 1 if M − 1 is a power of 2, otherwise n 2 . Two different combinatorial proofs of this assertion were found by the author. The first proof, which is written in the paper, is based on an existence theorem for k-uniform hypergraphs where the degrees of its vertices are limited by a given upper bound. The second proof, which is omitted for the sake of conciseness, is based on Baranyai's theorem on l-factorization of a complete k-uniform hypergraph.