Bounds for Complete Arcs in PG(3, q) and Covering Codes of Radius 3, Codimension 4, Under a Certain Probabilistic Conjecture

摘要

Let t(N, q) be the smallest size of a complete arc in the N-dimensional projective space PG(N, q) over the Galois field of order q. The d-length function ℓ _q(r,R,d) is the smallest length of a g-ary linear code of codimension (redundancy) r, covering radius R, and minimum distance d; in particular, ℓ _q(4, 3, 5) is the smallest length n of an [n, n -4, 5]_q3 quasi-perfect MDS code. By the definitions, ℓ _q(4, 3, 5) = t(3, q). In this paper, a step-by-step construction of complete arcs in PG(3, q) is considered. It is proved that uncovered points are uniformly distributed in the space. A natural conjecture on quantitative estimations of the construction is presented. Under this conjecture, new upper bounds on t(3, q) are obtained, in particular, t(3, q) < 2.93(q ln q)~(1/3).