Complete Caps in AG(N,q) with BothNandqOdd

Abstract

In this work, an inductive method to construct complete caps in affine spaces AG(N,q) is provided. Using this tool, for N≥5 odd and q odd, complete caps smaller than all already known ones are obtained.

Small complete caps in PG(4n+1,q)${\rm PG}(4n+1, q)$

“…Therefore 𝑥 2(𝑞+1) = (𝜌 − 𝜆𝑦 2 ) 𝑞+1 = (𝜌 − 𝜆𝑦 𝑞+1 ) 2 , that is, (𝑦 − 𝑦 𝑞 ) 2 = 0, a contradiction. □ Definition 1.…”

“…𝑟−1 2 . Constructions of complete caps whose size is close to this lower bound are only known for 𝑞 even.…”

“…In 1959 Segre [21] proved the existence of a complete cap of PG(3, 𝑞), 𝑞 even, of size 3𝑞 + 2. Later on, Pambianco and Storme [20], based on Segre's construction, proved the existence of complete caps of PG(𝑟, 𝑞), 𝑞 even, of size 𝑞 𝑟 2 + 3(𝑞 𝑟−2 2 + ⋯ + 𝑞) + 2, if 𝑟 is even, and 3(𝑞 𝑟−1 2 + ⋯ + 𝑞) + 2, if 𝑟 is odd. These ideas have been further developed by Giulietti [11] and Davydov, Giulietti, Marcugini and Pambianco [8] who exhibited smaller caps in PG(𝑟, 𝑞), 𝑞 even.…”

“…A probabilistic approach has been employed in [3,16] to provide an upper bound of roughly speaking 𝑞 𝑟−1 2 log 𝑐 𝑞 on 𝑡 2 (𝑟, 𝑞), for some positive constant 𝑐 independent from 𝑟 and 𝑞. For further or more recent results on this topic, see also [2,4,7,9]. For an account on the various constructive methods known so far the reader is referred to [12].…”

Small complete caps in ${\rm PG}(4n + 1, q)$

“…A probabilistic approach has been employed in [7] to provide an upper bound on t 2 (r, q). For further or more recent results on this topic see also [2,3,4,9]. For an account on the various constructive methods known so far the reader is referred to [12].…”