| Abstract: | Covering arrays for words of length  over a  ‐letter alphabet are  arrays with entries from the alphabet so that for each choice of  columns, each of the   ‐letter words appears at least once among the rows of the selected columns. We study two schemes in which all words are not considered to be different. In the first case known as partitioning hash families, words are equivalent if they induce the same partition of a  element set. In the second case, words of the same weight are equivalent. In both cases, we produce logarithmic upper bounds on the minimum size  of a covering array. Definitive results for  , as well as general results, are provided. |
