Linear codes with covering radius 3

Davydov, Alexander A.; Östergård, Patric R. J.

doi:10.1007/s10623-009-9322-y

Cited by 11 publications

(24 citation statements)

References 15 publications

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“…By [21, Th. 1], [27,Ths 4 and 9], there exists an infinite family of [n, n − r] 3 2 codes with the following parameters: Let q = 4. In [28] an infinite family of [n, n − r] 4 2 codes is obtained with parameters ,…”

Section: Infinite Code Families Of Odd Codimension R = 2t +mentioning

confidence: 99%

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Linear nonbinary covering codes and saturating sets in projective spaces

Davydov¹,

Giulietti²,

Marcugini³

et al. 2011

Advances in Mathematics of Communications

121

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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

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Section: Infinite Code Families Of Odd Codimension R = 2t +mentioning

confidence: 99%

“…R,q such that its supporting codes are [n u , n u − r u ] q R codes with codimension r u = Ru + γ and length n u = f R,q is a standard method of investigation of linear covering codes, see [3], [17], [18], [24], [27]- [29], [35] R,q , γ = 0, 1, . .…”

mentioning

confidence: 99%

Linear nonbinary covering codes and saturating sets in projective spaces

Davydov¹,

Giulietti²,

Marcugini³

et al. 2011

Advances in Mathematics of Communications

121

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“…The known results on t ( N , q ) and , , can be found in [ 1 – 6 , 13 , 26 ], see also the references therein. In [ 1 – 4 ], the case is considered.…”

Section: Introductionmentioning

confidence: 99%

“…In [ 6 , 26 ], algebraic constructions for are proposed. Computer search results for are given in [ 4 , 5 , 13 ].…”

Section: Introductionmentioning

confidence: 99%

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

Davydov

Marcugini

Pambianco

2020

Lecture Notes in Computer Science

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Let t ( N , q ) be the smallest size of a complete arc in the N -dimensional projective space over the Galois field of order q . The d -length function is the smallest length of a q -ary linear code of codimension (redundancy) r , covering radius R , and minimum distance d ; in particular, is the smallest length n of an quasi-perfect MDS code. By the definitions, . In this paper, a step-by-step construction of complete arcs in 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, .

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