Packing and Covering Properties of Rank Metric Codes

Gadouleau, Maximilien; Yan, Zhiyuan

doi:10.1109/tit.2008.928284

Cited by 40 publications

(56 citation statements)

References 26 publications

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“…Taking advantage of this extension field, we can then replace the linear MDS code in Ozarow-Wyner coset coding scheme by a linear maximum-rank-distance (MRD) code, which is essentially a linear code over F q m that is optimal for the rank metric. Codes in the rank metric were studied by a number of authors [9]- [11] and have been proposed for error control in random network coding [12], [13]. Here, we show that, since the channel to the eavesdropper is a linear transformation channel (rather than an erasure channel), rankmetric codes are naturally suitable to the problem (as opposed to classical codes designed for the Hamming metric).…”

Section: Introductionmentioning

confidence: 83%

Universal Secure Network Coding via Rank-Metric Codes

Silva

Kschischang

2011

IEEE Trans. Inform. Theory

143

166

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Abstract-The problem of securing a network coding communication system against an eavesdropper adversary is considered. The network implements linear network coding to deliver n packets from source to each receiver, and the adversary can eavesdrop on µ arbitrarily chosen links. The objective is to provide reliable communication to all receivers, while guaranteeing that the source information remains information-theoretically secure from the adversary. A coding scheme is proposed that can achieve the maximum possible rate of n − µ packets. The scheme, which is based on rank-metric codes, has the distinctive property of being universal: it can be applied on top of any communication network without requiring knowledge of or any modifications on the underlying network code. The only requirement of the scheme is that the packet length be at least n, which is shown to be strictly necessary for universal communication at the maximum rate. A further scenario is considered where the adversary is allowed not only to eavesdrop but also to inject up to t erroneous packets into the network, and the network may suffer from a rank deficiency of at most ρ. In this case, the proposed scheme can be extended to achieve the rate of n−ρ−2t−µ packets. This rate is shown to be optimal under the assumption of zero-error communication.

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Section: Introductionmentioning

confidence: 83%

Universal Secure Network Coding via Rank-Metric Codes

Silva

Kschischang

2011

IEEE Trans. Inform. Theory

143

166

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“…We comment that the function f 2 has connections with rank metric codes (see e.g. [8], [17] for example). For a fixed matrix X of rank r X , the function f 2 (r X , r B , r) gives the number of matrices B of rank r B such that rk(X + B) = r. This is equal to the number of matrices B of rank r B such that rk(X − B ) = r (setting B = −B).…”

Section: Matrices Over Finite Fieldsmentioning

confidence: 99%

Finite-Field Matrix Channels for Network Coding

Blackburn

Claridge

2019

IEEE Trans. Inform. Theory

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In 2010, Silva, Kschischang and Kötter studied certain classes of finite field matrix channels in order to model random linear network coding where exactly t random errors are introduced.In this paper we consider a generalisation of these matrix channels where the number of errors is not required to be constant, indeed the number of errors may follow any distribution. We show that a capacity-achieving input distribution can always be taken to have a very restricted form (the distribution should be uniform given the rank of the input matrix). This result complements, and is inspired by, a paper of Nobrega, Silva and Uchoa-Filho, that establishes a similar result for a class of matrix channels that model network coding with link erasures. Our result shows that the capacity of our channels can be expressed as a maximisation over probability distributions on the set of possible ranks of input matrices: a set of linear rather than exponential size.

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“…Rank metric codes have seen a recent resurgence of interest both for their potential use in code based cryptography and as error-correcting codes in network communications [19,26,27,36,39,40]. They are also intriguing as mathematical objects in their own right, and several researchers have sought to describe their structural properties [1,4,7,12,13,14,20,21,25,32,35]. However, the general theory of rank metric codes is still rather unexplored.…”

Section: Introductionmentioning

confidence: 99%

Partition-balanced families of codes and asymptotic enumeration in coding theory

Byrne

Ravagnani

2020

Journal of Combinatorial Theory, Series A

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We introduce the class of partition-balanced families of codes, and show how to exploit their combinatorial invariants to obtain upper and lower bounds on the number of codes that have a prescribed property. In particular, we derive precise asymptotic estimates on the density functions of several classes of codes that are extremal with respect to minimum distance, covering radius, and maximality. The techniques developed in this paper apply to various distance functions, including the Hamming and the rank metric distances. Applications of our results show that, unlike the Fqm -linear MRD codes, the Fq-linear MRD codes are not dense in the family of codes of the same dimension. More precisely, we show that the density of Fq-linear MRD codes in F n×m q in the set of all matrix codes of the same dimension is asymptotically at most 1/2, both as q → +∞ and as m → +∞. We also prove that MDS and Fqm -linear MRD codes are dense in the family of maximal codes. Although there does not exist a direct analogue of the redundancy bound for the covering radius of Fq-linear rank metric codes, we show that a similar bound is satisfied by a uniformly random matrix code with high probability. In particular, we prove that codes meeting this bound are dense. Finally, we compute the average weight distribution of linear codes in the rank metric, and other parameters that generalize the total weight of a linear code.2010 Mathematics Subject Classification. 05A16, 11T71.

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Packing and Covering Properties of Rank Metric Codes

Cited by 40 publications

References 26 publications

Universal Secure Network Coding via Rank-Metric Codes

Universal Secure Network Coding via Rank-Metric Codes

Finite-Field Matrix Channels for Network Coding

Partition-balanced families of codes and asymptotic enumeration in coding theory

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