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

被引：3

作者：

Davydov, Alexander A. ^{[1
]}

Marcugini, Stefano ^{[2
]}

Pambianco, Fernanda ^{[2
]}

机构：

[1] Russian Acad Sci, Inst Informat Transmiss Problems, Kharkevich Inst, Moscow, Russia

[2] Perugia Univ, Dept Math & Comp Sci, Perugia, Italy

来源：

COMPUTATIONAL SCIENCE AND ITS APPLICATIONS - ICCSA 2020, PT I | 2020年 / 12249卷

关键词：

Arcs in projective spaces; Covering codes; The d-length function; SMALLEST SIZE; SATURATING SETS; LINEAR CODES;

D O I：

10.1007/978-3-030-58799-4_8

中图分类号：

TP39 [计算机的应用];

学科分类号：

081203 ; 0835 ;

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 q-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](q)3 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 3 root q ln q.

引用

页码：107 / 122

页数：16

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