New LMRD Code Bounds for Constant Dimension Codes and Improved Constructions

Heinlein, Daniel

doi:10.1109/tit.2019.2905002

Cited by 14 publications

(11 citation statements)

References 26 publications

(37 reference statements)

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“…If a CDC contains an LMRD, see Section II for the definition, then the best known upper bound on the cardinality for the general case can be improved. Corresponding results have been obtained in [2], [4] for a restricted range of parameters. Here we remove the restriction and generalize those bounds to all parameters.…”

Section: Ksupporting

confidence: 63%

“…For some special cases, Etzion and Silberstein have demonstrated that one can obtain tighter upper bounds on the maximum possible cardinality of CDCs if we assume that an LMRD code is contained [2]. The range of applicable parameters was partially extended by Heinlein in [4]. Here we fully generalize those bounds, which also sheds some light on recent constructions.…”

Section: Generalized Lmrd Code Bounds For Constant Dimension Codesmentioning

confidence: 72%

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Generalized LMRD Code Bounds for Constant Dimension Codes

Kurz

2020

IEEE Commun. Lett.

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Section: Ksupporting

confidence: 63%

Section: Generalized Lmrd Code Bounds For Constant Dimension Codesmentioning

confidence: 72%

Generalized LMRD Code Bounds for Constant Dimension Codes

Kurz

2020

IEEE Commun. Lett.

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“…Lifted MRD codes can often be extended to larger subspace codes, but again not always to optimal subspace codes. Some recent results on this can be found in [15], [44], with further background and open problems in [32].…”

Section: Subspace Codesmentioning

confidence: 99%

13. MRD codes: constructions and connections

Sheekey¹

2019

Combinatorics and Finite Fields

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Rank-metric codes are codes consisting of matrices with entries in a finite field, with the distance between two matrices being the rank of their difference. Codes with maximum size for a fixed minimum distance are called Maximum Rank Distance (MRD) codes. Such codes were constructed and studied independently by Delsarte (1978), Gabidulin (1985), Roth (1991), andCooperstein (1998). Rank-metric codes have seen renewed interest in recent years due to their applications in random linear network coding.MRD codes also have interesting connections to other topics such as semifields (finite nonassociative division algebras), finite geometry, linearized polynomials, and cryptography. In this chapter we will survey the known constructions and applications of MRD codes, and present some open problems.The most well-known and widely-used example are codes in the Hamming metric; codewords are taken from a vector-space over a finite field, and distance between two vectors in F n q is defined as the number of positions in which they differ. We refer to [65] for a detailed reference on this well-studied topic.In rank-metric coding, codewords are instead taken from the set of matrices over a finite field, with the distance between two matrices defined as the rank of their difference.Definition 1. Let F q denote the finite field with q elements, and M n×m (F q ) the set of n × m matrices with entries in F q , with m ≤ n. The rank-weight of a matrix X is defined to be its usual column rank, and denoted by rank(X). The rank-distance between two matrices X, Y ∈ M n×m (F q ) is denoted by d(X, Y ) and defined asA rank-metric code is then a subset C of M n×m (F q ). The minimum distance of a set C with at least two elements, denoted d(C) is MRD codesA main goal in coding theory is to find codes of maximum possible size for a given minimum distance. While there are a variety of bounds for codes in the Hamming metric [65], for rank metric codes we need only the following simple analogue of the Singleton bound, first proved by Delsarte [24]. We include a short proof for illustrative purposes.Theorem 1 (Singleton-like bound, Delsarte). Suppose C ⊂ M n×m (F q ) is such that d(C) = d. Then |C| ≤ q n(m−d+1) .(1)Proof. Suppose |C| > q n(m−d+1) . By the pigeonhole principle, there must exist two distinct elements X, Y of C which coincide in every entry of their first m − d + 1 rows. Thus X − Y has at least m − d + 1 zero rows, and thus rank(X − Y ) ≤ d − 1 < d, a contradiction.is said to be a Maximum Rank Distance code, or MRD-code for short. We say C is a [n × m, d(C)]-MRD code.MRD-codes are the rank-metric analogue of MDS-codes in the Hamming metric. Unlike MDS codes, MRD codes in fact exist for all values of q, n, m, and all d. This was first shown by Delsarte [24]; indeed, he showed that the bound can be met by an additively-closed set.Theorem 2 ([24]). There exists an MRD code in M n×m (F q ) with minimum distance d for all q, n, m, and all d.This article with focus on the case where C is an MRD-code which is additively-closed, which we wi...

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“…and we can calculate that rank V V A l = 5 or 6, for any l ∈ {0, 1, • • • , 62}. Using (2), we get Therefore, C(3, 0, 6) is an (6,63,4,3) unitary cyclic orbit code.…”

Section: Constructions Of Cyclic Orbit Codes Base On Totally Isotropic Subspaces In Unitary Spacementioning

confidence: 99%

“…Xu and Chen [3] presented an effective construction which can be seen as a generalization of the lifted maximum rank distance codes. Heinlein [4] generalized the upper bounds of for constant dimension codes which contain lifted maximum rank distance codes. In [5], Luerssen et al proposed a new construction coming from Corollary 39 in [6] which was named as the linkage construction.…”

Section: Introductionmentioning

confidence: 99%

Constructions of Orbit Codes Based on Unitary Spaces Over Finite Fields

Chen

Qin

2021

IEEE Access

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Orbit codes, as special constant dimension codes, have attracted much attention due to their applications for error correction in random network coding. This paper is devoted to constructing large orbit codes by making full use of unitary space. Firstly, we construct a cyclic unitary group of order q 2n − 1 by means of the companion matrix of a primitive polynomial over finite fields F q 2 , and so the corresponding code is unitary cyclic orbit code. As a special application, a new quaternary orbit code (6, 63, 4, 3) is given. Secondly, we obtain orbit codes with large size using the external direct product of unitary groups acting on the direct sum of subspaces. Finally, a table is given for illustrating our codes improve upon those constructed by Trautmann et al. and Poroch et al. INDEX TERMSConstant dimension codes, orbit codes, unitary space,unitary group, primitive polynomials.

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New LMRD Code Bounds for Constant Dimension Codes and Improved Constructions

Abstract: We generalize upper bounds for constant dimension codes containing a lifted maximum rank distance code first studied by Etzion and Silberstein. The proof allows to construct several improved codes.

Cited by 14 publications

References 26 publications

Generalized LMRD Code Bounds for Constant Dimension Codes

Generalized LMRD Code Bounds for Constant Dimension Codes

13. MRD codes: constructions and connections

Constructions of Orbit Codes Based on Unitary Spaces Over Finite Fields

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