New Constructions of Subspace Codes Using Subsets of MRD codes in Several Blocks

Chen, Hao; He, Xianmang; Weng, Jian; Xu, Liqing

doi:10.48550/arxiv.1908.03804

Cited by 3 publications

(9 citation statements)

References 16 publications

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“…We remark that very recently, based on Theorem 2.1, Chen et.al [3] and Heinlein [15] generalized linkage constructions to establish some lower bounds of CDCs independently.…”

Section: Parallel Constructionmentioning

confidence: 99%

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Parallel multilevel constructions for constant dimension codes

Liu

Chang

2019

Preprint

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Constant dimension codes (CDCs), as special subspace codes, have received a lot of attention due to their application in random network coding. This paper introduces a family of new codes, called rank metric codes with given ranks (GRMCs), to generalize the parallel construction in [Xu and Chen, IEEE Trans. Inf. Theory, 64 (2018), 6315-6319] and the classic multilevel construction. A Singleton-like upper bound and a lower bound for GRMCs derived from Gabidulin codes are given. Via GRMCs, two effective constructions for CDCs are presented by combining the parallel construction and the multilevel construction. Many CDCs with larger size than the previously best known codes are given. The ratio between the new lower bound and the known upper bound for (4δ, 2δ, 2δ) q -CDCs is calculated. It is greater than 0.99926 for any prime power q and any δ ≥ 3.

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“…We remark that very recently, based on Theorem 2.1, Chen et.al [3] and Heinlein [15] generalized linkage constructions to establish some lower bounds of CDCs independently.…”

Section: Parallel Constructionmentioning

confidence: 99%

“…We always denote by γ i , 0 ≤ i ≤ n − 1, the number of dots in the i-th column of F. Given positive integers m and n, and 1 3,4,5], where…”

Section: Ferrers Diagram Rank-metric Codesmentioning

confidence: 99%

Parallel multilevel constructions for constant dimension codes

Liu

Chang

2019

Preprint

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“…Finally, Theorem 19 was improved by Chen, He, Weng, and Xu in [2] by allowing the dimensions of the ambient spaces to vary. This is the so-called parallel linkage construction.…”

Section: For 3 ≤ K ≤ S +mentioning

confidence: 99%

Generalized linkage construction for constant-dimension codes

Heinlein¹

2019

Preprint

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A constant-dimension code (CDC) is a set of subspaces of constant dimension in a common vector space with upper bounded pairwise intersection. We improve and generalize two constructions for CDCs, the improved linkage construction and the parallel linkage construction, to the generalized linkage construction which in turn yields many improved lower bounds for the cardinalities of CDCs; a quantity not known in general.

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“…One main problem for the constant dimension subspace coding is to determine the maximal possible size A q (n, d, k) of such a code for given parameters n, d, k, q. We refer to papers [9,7,8,22,11,15,24,1,14,20,19,3,18] and the nice webpage [12] for latest constructions and references. The presently known best constant dimension subspace codes for n ≤ 19, q ≤ 9 are listed in the table in the webpage [12].…”

Section: Introductionmentioning

confidence: 99%

“…Many presently best records of constant dimension subspace codes are from the Cossidente-Kurz-Marino-Pavese combining construction in [3]. Though there are various good constructions [24,1,14,20,19,3,18] since 2018, it seems that for many small parameter cases there are still big gaps between the presently known best upper bounds and lower bounds. This observation leads us to believe that there are some places in the present constructions which are needed to be filled with some new k-dimensional subspaces while the subspace distance can be preserved.…”

Section: Introductionmentioning

confidence: 99%

Parameter-controlled inserting constructions of constant dimension subspace codes

Lao¹,

Chen²,

Weng³

et al. 2020

Preprint

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A basic problem in constant dimension subspace coding is to determine the maximal possible size A q (n, d, k) of a set of k-dimensional subspaces in F n q such that the subspace distance satisfies dis(U, V ) = 2k − 2 dim(U ∩ V ) ≥ d for any two different k-dimensional subspaces U and V in this set. In this paper we propose new parameter-controlled inserting constructions of constant dimension subspace codes. These inserting constructions are flexible because they are controlled by parameters. Several new better lower bounds which are better than all previously constructive lower bounds can be derived from our flexible inserting constructions. 141 new constant dimension subspace codes of distances 4, 6, 8 better than previously best known codes are constructed.

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New Constructions of Subspace Codes Using Subsets of MRD codes in Several Blocks

Cited by 3 publications

References 16 publications

Parallel multilevel constructions for constant dimension codes

Parallel multilevel constructions for constant dimension codes

Generalized linkage construction for constant-dimension codes

Parameter-controlled inserting constructions of constant dimension subspace codes

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