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New upper bounds on the smallest size t_{2}(2,q) of a complete arc in the projective plane PG(2,q) are obtained for 853≤q≤2879 and q=3511,4096, 4523,5003,5347,5641,5843,6011. For q≤2377, the smallest size t_{2}(2,q) is 7.
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