Decoding of random network codes

Computing Research Repository, 2007

It is shown that the error control problem in random network coding can be reformulated as a generalized decoding problem for rank-metric codes. This result allows many of the tools developed for rank-metric codes to be applied to random network coding. In the generalized decoding problem induced by random network coding, the channel may supply partial information about the error in the form of erasures (knowledge of an error location but not its value) and deviations (knowledge of an error value but not its location). For Gabidulin codes, an important family of maximum rank distance codes, an efficient decoding algorithm is proposed that can fully exploit the correction capability of the code; namely, it can correct any pattern of errors, µ erasures and δ deviations provided

2016

In this paper is we establish the role of Rank-metric Codes for error correction in Random network Coding. For this, first we try to understand the process of communication and its components. Thereafter, describing the problem in achieving error free communication and making necessary assumptions, we arrive at the conclusion.

2009 Workshop on Network Coding, Theory, and Applications, 2009

The random network coding approach is an effective technique for linear network coding, however it is highly susceptible to errors and adversarial attacks. Recently Kötter and Kschischang [14] introduced the operator channel, where the inputs and outputs are subspaces of a given vector space, showing that this is a natural transmission model in noncoherent random network coding. A suitable metric, defined for subspaces:

2010

The multiplicative-additive finite-field matrix channel arises as an adequate model for linear network coding systems when links are subject to errors and erasures, and both the network topology and the network code are unknown. In a previous work we proposed a general construction of multishot codes for this channel based on the multilevel coding theory. Herein we apply this construction to the rank-metric space, obtaining multishot rank-metric codes which, by lifting, can be converted to codes for the aforementioned channel. We also adapt well-known encoding and decoding algorithms to the considered situation.

Designs, Codes and Cryptography, 2008

In this paper, transmitted signals are considered as square matrices of the Maximum rank distance (MRD) (n, k, d)-codes. A new composed decoding algorithm is proposed to correct simultaneously rank errors and rank erasures. If the rank of errors and erasures is not greater than the Singleton bound, then the algorithm gives always the correct decision. If it is not a case, then the algorithm gives still the correct solution in many cases but some times the unique solution may not exist.

IEEE Transactions on Information Theory, 2015

By extending the notion of minimum rank distance, this paper introduces two new relative code parameters of a linear code C 1 of length n over a field extension F q m and its subcode C 2 C 1 . One is called the relative dimension/intersection profile (RDIP), and the other is called the relative generalized rank weight (RGRW). We clarify their basic properties and the relation between the RGRW and the minimum rank distance. As applications of the RDIP and the RGRW, the security performance and the error correction capability of secure network coding, guaranteed independently of the underlying network code, are analyzed and clarified. We propose a construction of secure network coding scheme, and analyze its security performance and error correction capability as an example of applications of the RDIP and the RGRW. Silva and Kschischang showed the existence of a secure network coding in which no part of the secret message is revealed to the adversary even if any dim C 1 -1 links are wiretapped, which is guaranteed over any underlying network code. However, the explicit construction of such a scheme remained an open problem. Our new construction is just one instance of secure network coding that solves this open problem.

2007 10th Canadian Workshop on Information Theory (CWIT), 2007

The idea of priority encoding transmission (PET), although very useful to provide graceful degradation of performance in the presence of packet loss, presents a challenge when one tries to apply it in a random network coding environment. So far, the only solution proposed relies on the rateless feature of network coding and is therefore not suitable for delay-constrained applications. In this paper, a PET system based on rank-metric codes is proposed. This system can provide strict combinatorial guarantees of recovery. The system is also able to provide forward error correction of corrupt packets that could potentially be introduced by malicious users. Application of the system to streaming media broadcasting is proposed, illustrating the potential practical utility of this approach.

2021 XVII International Symposium "Problems of Redundancy in Information and Control Systems" (REDUNDANCY)

In this paper an interpolation-based decoding algorithm to decode Gabidulin codes transmitted through a new communication model is proposed. The algorithm is able to decode rank errors beyond half the minimum distance by one unit. Also the existing decoding algorithms for generalized twisted Gabidulin codes and additive generalized twisted Gabidulin codes are improved.

2017

In 2000, Ahlswede, Cai and Li introduced network coding, a technique used to improve the efficiency of information flow through networks by allowing intermediate nodes to compute with and modify data. In practice random linear network coding is used, where the nodes transmit random linear combinations of their incoming packets. This thesis is concerned with several mathematical problems motivated by network coding. We first consider partial decoding in random linear network coding. By noting the equivalence to an enumeration problem in linear algebra, we compute the exact probability of a receiver decoding a fraction of the source message. We investigate the consequences when using both systematic and non-systematic network coding. We then consider mathematical models for network coding. 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. We introduce a gene...

ArXiv, 2022

The rank metric measures the distance between two matrices by the rank of their difference. Codes designed for the rank metric have attracted considerable attention in recent years, reinforced by network coding and further motivated by a variety of applications. In code-based cryptography, the hardness of the corresponding generic decoding problem can lead to systems with reduced public-key size. In distributed data storage, codes in the rank metric have been used repeatedly to construct codes with locality, and in coded caching, they have been employed for the placement of coded symbols. This survey gives a general introduction to rank-metric codes, explains their most important applications, and highlights their relevance to these areas of research.