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

Abstract

Let t(N, q) be the smallest size of a complete arc in the N-dimensional projective space \(\mathrm {PG}(N,q)\) over the Galois field of order q. The d-length function \(\ell _q(r,R,d)\) is the smallest length of a q-ary linear code of codimension (redundancy) r, covering radius R, and minimum distance d; in particular, \(\ell _q(4,3,5)\) is the smallest length n of an \([n,n-4,5]_q3\) quasi-perfect MDS code. By the definitions, \(\ell _q(4,3,5)=t(3,q)\). In this paper, a step-by-step construction of complete arcs in \(\mathrm {PG}(3,q)\) 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, \(t(3,q)<2.93\root 3 \of {q\ln q}\).