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

Davydov, Alexander A.; Marcugini, Stefano; Pambianco, Fernanda

doi:10.1007/s10623-019-00649-2

Cited by 17 publications

(23 citation statements)

References 20 publications

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“…In this case l q (5, 3) < 2.884q 2 3 (lnq) 1 3 was proved, see [5]. More generally the following result follows from the bound in [14].…”

Section: For a Code Family Of Lengthmentioning

confidence: 76%

“…Actually this can be proved directly from the the characterization of the covering radius of linear codes from its parity check matrix. There are a lot of results about this topic, we refer to [13,14,5] and references therein. From our covering code upper bounds the size of an (R, L) list-decodable code C of length n satisfying n ≥ l q (r, R) has to satisfy…”

Section: For a Code Family Of Lengthmentioning

confidence: 99%

“…Actually the covering radius R covering (C) of a linear [n, k] q code C ⊂ F n q can be determined as follows. If H is any (n − k) × n parity check matrix of C, R covering (C) is the least integer such that every vector in F n−k q can be represented as F q linear combinations of R covering (C) or fewer columns of H, see [57,14]. It follows the redundancy upper bound R covering (C) ≤ n − k for a linear [n, k] q code, see [11].…”

Section: Introductionmentioning

confidence: 99%

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List-decodable Codes and Covering Codes

Chen¹

2021

Preprint

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The list-decodable code has been an active topic in theoretical computer science since the seminal papers of M. Sudan and V. Guruswami in 1997-1998. There are general results about the list-decodability to the Johnson radius and the list-decoding capacity theorem. However few results about general constraints on rates, list-decodable radius and list sizes for list-decodable codes have been obtained. List-decodable codes are also considered in rank-metric, subspace metric, cover-metric, pair metric and insdel metric settings. In this paper we show that rates, list-decodable radius and list sizes are closely related to the classical topic of covering codes. We prove new general simple but strong upper bounds for list-decodable codes in general finite metric spaces based on various covering codes. The general covering code upper bounds can be applied to the case that the volumes of the balls depend on the centers, not only on the radius. Then any good upper bound on the covering radius or the size of covering code imply a good upper bound on the sizes of list-decodable codes. Hence the list-decodablity of codes is a strong constraint from the view of covering codes on general finite metric spaces. Our results give exponential improvements on the recent generalized Singleton upper bound of Shangguan and Tamo in STOC 2020 for Hamming metric list-decodable codes, when the code lengths are large. The generalized Singleton upper bound on the average-radius list-decodable codes is also given from our general covering code upper bound. The asymptotic forms of covering code bounds can partially recover the Blinovsky bound and the combinatorial bound of Guruswami-Håstad-Sudan-Zuckerman in Hamming metric setting. We also suggest to study the combinatorial covering list-decodable codes as a natural generalization of combinatorial listdecodable codes. We apply our general covering code upper bounds for

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“…In this case l q (5, 3) < 2.884q 2 3 (lnq) 1 3 was proved, see [5]. More generally the following result follows from the bound in [14].…”

Section: For a Code Family Of Lengthmentioning

confidence: 76%

Section: For a Code Family Of Lengthmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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List-decodable Codes and Covering Codes

Chen¹

2021

Preprint

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“…Recently, improvements on the upper bound on the minimum size of a ρ-saturating set have been obtained in [17,18]. Theorem 4.3 ([18]).…”

Section: Theorem 42 ([16]mentioning

confidence: 99%

Small Strong Blocking Sets by Concatenation

Bartoli¹,

Borello²

2021

Preprint

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Strong blocking sets and their counterparts, minimal codes, attracted lots of attention in the last years. Combining the concatenating construction of codes with a geometric insight into the minimality condition, we explicitly provide infinite families of small strong blocking sets, whose size is linear in the dimension of the ambient projective spaces. As a byproduct, small saturating sets are obtained.

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“…The length function l q (r, R) is the smallest length of a q-ary linear code of co-dimension r and covering radius R. There is a one-to-one correspondence between [n, n − r] q R codes and (R − 1)-saturating sets of size n in PG(r − 1, q). This implies l q (r, R) = s q (r − 1, R − 1) [17,18]. Applying a random construction based on point sets of subspaces in the spirit of higgledy-piggledy line sets (i.e.…”

Section: Introductionmentioning

confidence: 99%

Short minimal codes and covering codes via strong blocking sets in projective spaces

Héger¹,

Nagy²

2021

Preprint

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Minimal linear codes are in one-to-one correspondence with special types of blocking sets of projective spaces over a finite field, which are called strong or cutting blocking sets. In this paper we prove an upper bound on the minimal length of minimal codes of dimension k over the q-element Galois field which is linear in both q and k, hence improve the previous superlinear bounds. This result determines the minimal length up to a small constant factor. We also improve the lower and upper bounds on the size of so called higgledy-piggledy line sets in projective spaces and apply these results to present improved bounds on the size of covering codes and saturating sets in projective spaces as well. The contributions rely on geometric and probabilistic arguments.

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

Cited by 17 publications

References 20 publications

List-decodable Codes and Covering Codes

List-decodable Codes and Covering Codes

Small Strong Blocking Sets by Concatenation

Short minimal codes and covering codes via strong blocking sets in projective spaces

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