Small complete caps inPG(N,q),qeven

Giulietti, Massimo

doi:10.1002/jcd.20131

Cited by 25 publications

(31 citation statements)

References 19 publications

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“…If q is even and N is odd, such bound is substantially sharp; see [20] for N = 3 and [6,12,15] for larger N . Otherwise, all known infinite families of complete caps have size far from (1.1); see the survey papers [13,17].…”

Section: Introductionmentioning

confidence: 99%

Complete caps in AG(3, q) from elliptic curves

Platoni

2014

J. Algebra Appl.

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Communicated by X. D. HouIn a three-dimensional Galois space of odd order q, the known infinite families of complete caps have size far from the theoretical lower bounds. In this paper, we investigate some caps defined from elliptic curves. In particular, we show that for each q between 100 and 350 they can be extended to complete caps, which turn out to be the smallest complete caps known in the literature.

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Section: Introductionmentioning

confidence: 99%

Complete caps in AG(3, q) from elliptic curves

Platoni

2014

J. Algebra Appl.

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“…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]…”

Section: Introduction the Main Resultsmentioning

confidence: 99%

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

et al. 2015

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In the projective plane PG(2, q), upper bounds on the smallest size t2(2, q) of a complete arc are considered. For a wide region of values of q, the results of computer search obtained and collected in the previous works of the authors and in the present paper are investigated. For q ≤ 301813, the search is complete in the sense that all prime powers are considered. This proves new upper bounds on t2(2, q) valid in this region, in particular t2(2, q) < 0.998 3q ln q for 7 ≤ q ≤ 160001;t2(2, q) < 1.05 3q ln q for 7 ≤ q ≤ 301813;t2(2, q) < √ q ln 0.7295 q for 109 ≤ q ≤ 160001;t2(2, q) < √ q ln 0.7404 q for 160001 < q ≤ 301813.The new upper bounds are obtained by finding new small complete arcs in PG(2, q) with the help of a computer search using randomized greedy algorithms and algorithms with fixed (lexicographical) order of points (FOP). Also, a number of sporadic q's with q ≤ 430007 is considered. Our investigations and results allow to conjecture that the 2-nd and 3-rd bounds above hold for all q ≥ 109. Finally, random complete arcs in PG(2, q), q ≤ 46337, q prime, are considered. The random complete arcs

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“…For the size t 2 (AG (N , q)) of the smallest complete cap in AG(N , q), the trivial lower bound is t 2 (AG(N , q)) > √ 2q N −1 2 . Unlike the even order case, where for every dimension N ≥ 3 there exist complete caps in AG(N , q) with less than q N 2 points ( [9,10,16,17], see Remark 1.5), for q odd complete k-caps in AG(N , q) with k ≤ q ( [2,6,7,8,15], see Remark 1.5). The aim of this paper is to describe small complete caps in AG(N , q) with q odd and N ≡ 0 (mod 4).…”

Section: Introductionmentioning

confidence: 91%

“…Therefore such caps can be viewed as complete caps in AG(N , q). Also, some known constructions of infinite families of complete caps in PG(N , q) are based on a complete cap K in an affine space PG(N , q) \ H ∞ , to which some properly chosen points on H ∞ are added (see [8,10,16,17]; note that in [8,16,17] the completeness of K in the affine space is proven without being explicitly stated). Results on t 2 (AG(N , q)) that can be deduced from [8,10,16,17] Throughout this section, we assume that q is an odd prime power and that N is even.…”

Section: Remark 15mentioning

confidence: 99%