Construction of Complete (k;r)-Arcs from Orbits in PG(3,11)

Abstract

The aim of this research is to partition into orbits using the subgroups of which are determined by the nontrivial positive divisors of the order of . These orbits were also studied from the perspective of arcs by finding complete and incomplete arcs.

Extension of Cap by Size and Degree in the Space PG(3,11)‎

“…) which is called the companion matrix over 𝐹 11 . The points of the finite projective space Σ 11 = 𝑃𝐺 (3,11) are constructing using 𝐶 𝑓 in the formula 𝑃(𝑖) ≔ [1,0,0,0]𝐶 𝑓 𝑖−1 , 𝑖 = 1,2, … ,1464, where 1464 = 𝜃(3,11) = 11 3 + 11 2 + 11 + 1 is the order of space. This matrix is cyclic of order 𝜃 (3,11), so to refer to the space's points we can use numeral form 𝑖 ≔ 𝑃(𝑖), 𝑖 = 1, … 𝜃 (3,11).…”

“…The points of the finite projective space Σ 11 = 𝑃𝐺 (3,11) are constructing using 𝐶 𝑓 in the formula 𝑃(𝑖) ≔ [1,0,0,0]𝐶 𝑓 𝑖−1 , 𝑖 = 1,2, … ,1464, where 1464 = 𝜃(3,11) = 11 3 + 11 2 + 11 + 1 is the order of space. This matrix is cyclic of order 𝜃 (3,11), so to refer to the space's points we can use numeral form 𝑖 ≔ 𝑃(𝑖), 𝑖 = 1, … 𝜃 (3,11). The lines in the space are found using the following formula: Let 𝑋 = [𝑥 0 , 𝑥 1 , 𝑥 2 , 𝑥 3 ] and 𝑌 = [𝑦 0 , 𝑦 1 , 𝑦 2 , 𝑦 3 ] be two point in Σ 11 on a line 𝑙.…”

“…The matrix 𝐶 𝑓 is belong to Σ 11 , so the 𝐻 = 〈𝐶 𝑓 〉 is cyclic subgroup of Caps from subgroups 〈𝑇 𝑡 〉 action on 𝑃𝐺(3,11) …”

“…is used to implement the above steps. Let 𝑌 = {2,3,4,6,8,12,24,61,122,183,244,366,488,732} be the divisors of 𝜃(3,11).2. Extension of CapsTheorem 2.1: Let 𝑂 𝑖 [𝑖, 𝑡] be the (𝑖, 𝑟)-caps as in Table 1, 𝑖, 𝑡 in 𝑌 and 𝑟 ≠ 12.…”