On Small Complete Arcs and Transitive A5-Invariant Arcs in the Projective Plane PG(2,q)

Abstract

Let q be an odd prime power such that q is a power of 5 or q≡±1 (mod 10). In this case, the projective plane PG(2,q) admits a collineation group G isomorphic to the alternating group A5. Transitive G‐invariant 30‐arcs are shown to exist for every q≥41. The completeness is also investigated, and complete 30‐arcs are found for q=109,121,125. Surprisingly, they are the smallest known complete arcs in the planes PG(2,109),PG(2,121), and PG(2,125). Moreover, computational results are presented for the cases G≅A4 and G≅S4. New upper bounds on the size of the smallest complete arc are obtained for q=67,97,137,139,151.

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

“…An open problem is to find small 1-saturating sets (respectively, short covering codes). A complete arc in PG(2, q) is, in particular, a 1-saturating set; often the smallest known complete arc is the smallest known 1-saturating set [17,27,28,38,62]. Let 1 (2, q) be the smallest size of a 1-saturating set in PG (2, q).…”

“…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]…”

“…For q ≤ 151, a number of improvements of t 2 (2, q), in comparison with [10,12], are given in [62] and cited in [4].…”

“…The most of the smallest known complete arcs for q ∈ Q used in the present paper are obtained by computer search based on randomized greedy algorithms [4,10,12,15,23,24,29], with the exception of (relatively few) theoretically constructed complete arcs, see, for example, [25,26,30,35,39,40,62] and references therein.…”

On the existence of PD-sets: Algorithms arising from automorphism groups of codes

“…This approach produced several examples of small PD-sets that, in some cases, can be proven to be the smallest possible. In general, it is convenient to start with a linear code with a large prescribed automorphism group and, not only because of permutation decoding, these codes have been widely investigated in the last years, see [2,4,6,9,8,10,11,12,13,14,18,21,24,36,37,38,39,40,41,49,50,51,52,54,56] and references therein. This paper deals with Problem (1).…”