Journal of Combinatorial Mathematics and Combinatorial Computing, 60 (2007), 7-32.
Let m2(N,q) denote the size of the largest caps in PG(N,q) and let m'2(N,q) denote the size of the second largest complete caps in PG(N,q). Presently, it is known that m2(4,5) = 111 and that m2(4,7) = 316. Via computer searches for caps in PG(4,5) using the result of Abatangelo, Larato and Korchmaros that m'2(3,5)=20, we improve the first upper bound to m2(4,5) = 88. Computer searches in PG(3,7) show that m'2(3,7)=32 and this latter result then improves the upper bound on m2(4,7) to m2(4,7) = 238. We also present the known upper bounds on m2(N,5) and m2(N,7) for N>4.
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