Small complete caps in Galois affine spaces

Abstract

Some new families of caps in Galois affine spaces AG(N , q) of dimension N ≡ 0 (mod 4) and odd order q are constructed. Such caps are proven to be complete by using some new ideas depending on the concept of a regular point with respect to a complete plane arc. As a corollary, an improvement on the currently known upper bounds on the size of the smallest complete caps in AG(N , q) is obtained.

Upper bounds on the smallest size of a complete arc in a finite Desarguesian projective plane based on computer search

“…Problems connected with small complete arcs in PG (2, q) are considered in [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]20,[23][24][25][26][27][28][30][31][32][33][34][35][36][37][38][39][40][41][42][44][45][46][47][48]50,51,[54][55][56][57][58][61][62][63]…”

Small complete caps from singular cubics, II

“…by (12), this can only happen when both e = 3 and g = (m − 1)/2 hold. This is impossible as (m, 6) = 1 is assumed.…”

“…Points in AG(N , q) can be identified with vectors of F q × F q × F q × F q . A key tool in this paper is the following result from [12].…”

“…Otherwise, all known infinite families of complete caps have size far from (1); see the survey papers [17,18] and the more recent works [1,4,5,7,8,[12][13][14]. For q odd and N = 2, the smallest explicit constructions go back to the late 80's, when Szőnyi described complete plane arcs of size approximately (q − 1)/m for any divisor m of q − 1 smaller than 1 C q 1/4 , with C a constant independent of q and greater than 1 [27,28] 1 .…”

On cyclic caps in 4-dimensional projective spaces

“…For q odd, some other constructions of complete caps of size approximately 2q 2 are available in the literature; see [11,24]. It should be noted, however, that the caps C 2(q 2 +1) are the largest complete cyclic caps in P G(4, q) constructed up to now for q ≡ 3 (mod 4).…”