Bicovering arcs and small complete caps from elliptic curves

Abstract

Bicovering arcs in Galois affine planes of odd order are a powerful tool for the construction of complete caps in spaces of arbitrarily higher dimensions. The aim of this paper is to investigate whether the arcs contained in elliptic cubic curves are bicovering. As a result, bicovering k-arcs in AG(2, q) of size k ≤ q/3 are obtained, provided that q − 1 has a prime divisor m with 7 < m < (1/8)q 1/4 . Such arcs produce complete caps of size kq (N −2)/2 in affine spaces of dimension N ≡ 0 (mod 4). When q = p h with p prime and h ≤ 8, these caps are the smallest known complete caps in AG(N, q), N ≡ 0 (mod 4).

Small complete caps from singular cubics, II

“…Let 2 s ≥ 4 be the highest power of 2 which divides 4 √ q − 1, and similarly 3 k ≥ 3 the highest power of 3 which divides 4 √ q − 1. Then, it is easy to see that one can choose m 1 and m 2 so that …”

“…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 .…”

“…It was first noted by Zirilli [31] that a non-trivial coset A of the group of the non-singular F q -rational points of X is a plane arc, provided that the index m of A is not divisible by 3. Since then, plane arcs in cubics have been thoroughly investigated, and complete caps have been obtained by recursive constructions from these arcs; see [1,4,9,11,19,[26][27][28][29][30].…”

Complete Caps in AG(N,q) with BothNandqOdd

“…For q odd and , , Davydov and Östergård constructed complete caps in with points. For q odd and complete caps of size were constructed in , whereas [, Theorem 2] yields the existence of complete caps in , , of size of the same order of magnitude as with , provided that has a prime divisor m greater than 7 and smaller than .…”

Small Complete Caps from Singular Cubics

“…We remark that bicovering properties of arcs contained in non-singular cubic curves are studied in [1], where complete caps in AG(N, q), N ≡ 0 (mod 4), of size less than 1 3 q N/2 are obtained under the assumption that q − 1 admits a prime divisor m with 7 < m < 1 8 q 1/4 . Some computer-assisted constructions of both bicovering and almost bicovering arcs in AG (2, q) for small q's are provided in [10].…”