Journal of Combinatorial Mathematics and Combinatorial Computing, 60 (2007), 7-32.
Abstract:
Let m_{2}(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 m_{2}(4,5) ‹= 111 and that m_{2}(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 m_{2}(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 m_{2}(4,7) to m_{2}(4,7) ‹= 238. We also present the known upper bounds on m_{2}(N,5) and m_{2}(N,7) for N>4.
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