New bounds for linear codes of covering radii 2 and 3

Bartoli, Daniele; Davydov, Alexander A.; Giulietti, Massimo; Marcugini, Stefano; Pambianco, Fernanda

doi:10.1007/s12095-018-0335-0

Cited by 8 publications

(19 citation statements)

References 18 publications

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“…The covering quality of a code is better if its covering density is smaller. Codes providing a convenient covering density are called covering codes [ 4 , 5 , 7 , 8 , 10 – 14 , 16 , 22 , 23 ]. Covering codes are applied, e.g.…”

Section: Introductionmentioning

confidence: 99%

“…Columns of a parity check matrix of an maximum distance separable (MDS) code can be considered as points (in homogeneous coordinates) of a complete n -arc in [ 4 – 6 , 10 , 14 , 16 , 18 , 20 , 22 ]. If these codes are quasi-perfect.…”

Section: Introductionmentioning

confidence: 99%

“…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. In [ 6 , 26 ], algebraic constructions for are proposed.…”

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%

“…In particular, the smallest known complete arcs obtained by algebraic constructions have size of order [ 6 , 26 ] while the bounds of this paper for are of order . Also, in this paper, the computer search is made in a more wide region of q than in [ 4 , 5 , 13 ]; many results of these papers are improved.…”

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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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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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New covering codes of radius R, codimension tR and $$tR+\frac{R}{2}$$, and saturating sets in projective spaces

Davydov

Marcugini

Pambianco

2019

Des. Codes Cryptogr.

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Constructing saturating sets in projective spaces using subgeometries

Denaux

2021

Des. Codes Cryptogr.

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A -saturating set of PG(N, q) is a point set S such that any point of PG(N, q) lies in a subspace of dimension at most spanned by points of S. It is generally known that a -saturating set of PG(N, q) has size at least c • q N − +1 , with c > 1 3 a constant. Our main result is the discovery of a -saturating set of size roughly ( +1)( +2) 2 q N − +1 if q = (q ) +1 , with q an arbitrary prime power. The existence of such a set improves most known upper bounds on the smallest possible size of -saturating sets if < 2N −1 3 . As saturating sets have a one-to-one correspondence to linear covering codes, this result improves existing upper bounds on the length and covering density of such codes.To prove that this construction is a -saturating set, we observe that the affine parts of q -subgeometries of PG(N, q) having a hyperplane in common, behave as certain lines of AG + 1, (q ) N . More precisely, these affine lines are the lines of the linear representation of a q -subgeometry PG( , q ) embedded in PG + 1, (q ) N .

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New bounds for linear codes of covering radii 2 and 3

Cited by 8 publications

References 18 publications

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

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

New covering codes of radius R, codimension tR and $$tR+\frac{R}{2}$$, and saturating sets in projective spaces

Constructing saturating sets in projective spaces using subgeometries

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