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A packing array is a b × k array, A with entriesai,j from a g-ary alphabet such that given any two columns,i and j, and for all ordered pairs of elements from a g-ary alphabet,(g1, g2), there is at most one row, r, such thatar,i = g1 andar,j = g2. Further, there is a set of at leastn rows that pairwise differ in each column: they are disjoint. A central question is to determine, forgiven g, k and n, the maximum possible b. We examine the implications whenn is close to g. We give a brief analysis of the case n = g and showthat 2g rows is always achievable whenever more than g exist. We give an upper bound derivedfrom design packing numbers when n = g − 1. When g + 1 ≤k then this bound is always at least as good as the modified Plotkin bound of [12]. When theassociated packing has as many points as blocks and has reasonably uniform replication numbers, we show thatthis bound is tight. In particular, finite geometries imply the existence of a family of optimal or near optimalpacking arrays. When no projective plane exists we present similarly strong results. This article completelydetermines the packing numbers, D(v, k, 1), when v .