Linear nonbinary covering codes and saturating sets in projective spaces

Abstract

Let AR,q denote a family of covering codes, in which the covering radius R and the size q of the underlying Galois field are fixed, while the code length tends to infinity. The construction of families with small asymptotic covering densities is a classical problem in the area Covering Codes.In this paper, infinite sets of families AR,q, where R is fixed but q ranges over an infinite set of prime powers are considered, and the dependence on q of the asymptotic covering densities of AR,q is investigated. It turns out that for the upper limit µ * q (R, AR,q) of the covering density of AR,q, the best possibility is(1)The main achievement of the present paper is the construction of optimal infinite sets of families AR,q, that is, sets of families such that (1) holds, for any covering radius R ≥ 2. We first showed that for a given R, to obtain optimal infinite sets of families it is enough to construct R infinite families A (0) R,q , A (1) R,q , . . . , A (R−1) R,q

New upper bounds on the smallest size of a saturating set in a projective plane

“…In further we use the results of [13,16] that give the following inductive construction. Then, under condition q + 1 ≥ 2n q , there is an infinite family of [n, n − r] q 2 codes with r = 2t − 1 ≥ 5, r = 9, 13, n = n q q t−2 + 2q t−3 , where t = 3, 4, 6, and t ≥ 8.…”

“…Surveys of the known [n, n − r] q 2 codes and saturating sets in PG(N, q) with N even can be found in [13,16,23]. In many cases bounds (3.1), (3.2) is better than the known ones.…”

“…Constructions of saturating sets (or the corresponding covering codes) whose size is close to this lower bound are only known for N odd, see [13,16,23] for survey. In particular, in [19,Theorem 9], see also [16,Section 4.3], the following bound is obtained by algebraic constructions:…”

“…where t = 2, 3, 5, and t ≥ 7. From (1.8), by using inductive constructions from [13,16], we obtained upper bounds on the smallest size of a saturating set in the N-dimensional projective space PG(N, q) with N even; see Section 3. In many cases these bounds are better than the known ones.…”

Double blocking sets of size 3q−1 in PG(2,q)Csajbók1, Héger2 2019European Journal of Combinatorics80220View full textAdd to dashboardCite

“…No other general results are known when t is a constant. (For particular results on double blocking sets, see [8] and [16] as cited in [4, p52].) If q is a prime, the situation is even worse.…”