Small Complete Caps from Singular Cubics

Small complete caps from singular cubics, II

“…Complete caps from singular cubics with a cusp were recently constructed by the same authors in [1]. The present paper can be considered a sequel of [1], in the sense that here, we deal with the two other projectively distinct singular cubics. The complete caps obtained here are significantly smaller than those constructed in [1], which have size roughly 2 p β q (4N −1)/8 , with β ∈ [1/8, 1] (see [1,Theorem 6.2]).…”

“…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].…”

“…[6,16]. Complete caps from singular cubics with a cusp were recently constructed by the same authors in [1]. The present paper can be considered a sequel of [1], in the sense that here, we deal with the two other projectively distinct singular cubics.…”

“…The present paper can be considered a sequel of [1], in the sense that here, we deal with the two other projectively distinct singular cubics. The complete caps obtained here are significantly smaller than those constructed in [1], which have size roughly 2 p β q (4N −1)/8 , with β ∈ [1/8, 1] (see [1,Theorem 6.2]). Also, the proofs here rely on deeper concepts from the theory of Function Fields and need original techniques, especially for the case of a cubic with an isolated double point.…”