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21
Construction of large constant dimension codes with a prescribed minimum distance
, 2008
"... In this paper we construct constant dimension space codes with prescribed minimum distance. There is an increased interest in space codes since a paper [13] by Kötter and Kschischang were they gave an application in network coding. There is also a connection to the theory of designs over finite fi ..."
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Cited by 46 (4 self)
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In this paper we construct constant dimension space codes with prescribed minimum distance. There is an increased interest in space codes since a paper [13] by Kötter and Kschischang were they gave an application in network coding. There is also a connection to the theory of designs over finite fields. We will modify a method of Braun, Kerber and Laue [7] which they used for the construction of designs over finite fields to do the construction of space codes. Using this approach we found many new constant dimension spaces codes with a larger number of codewords than previously known codes. We will finally give a table of the best found constant dimension space codes.
Recursive Code Construction for Random Networks
, 2008
"... We present a modification of KöetterKschischang codes for random networks. The new codes have higher information rate, while maintaining the same errorcorrecting capabilities. An efficient errorcorrecting algorithm is presented for these codes. ..."
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Cited by 30 (1 self)
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We present a modification of KöetterKschischang codes for random networks. The new codes have higher information rate, while maintaining the same errorcorrecting capabilities. An efficient errorcorrecting algorithm is presented for these codes.
Large constant dimension codes and lexicodes
 Advances in Mathematics of Communications
, 2011
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Bounds on List Decoding of RankMetric Codes
, 2013
"... So far, there is no polynomialtime list decoding algorithm (beyond half the minimum distance) for Gabidulin codes. These codes can be seen as the rankmetric equivalent of Reed–Solomon codes. In this paper, we provide bounds on the list size of rankmetric codes in order to understand whether poly ..."
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Cited by 4 (2 self)
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So far, there is no polynomialtime list decoding algorithm (beyond half the minimum distance) for Gabidulin codes. These codes can be seen as the rankmetric equivalent of Reed–Solomon codes. In this paper, we provide bounds on the list size of rankmetric codes in order to understand whether polynomialtime list decoding is possible or whether it works only with exponential time complexity. Three bounds on the list size are proven. The first one is a lower exponential bound for Gabidulin codes and shows that for these codes no polynomialtime list decoding beyond the Johnson radius exists. Second, an exponential upper bound is derived, which holds for any rankmetric code of length n and minimum rank distance d. The third bound proves that there exists a rankmetric code over Fqm of length n ≤ m such that the list size is exponential in the length for any radius greater than half the minimum rank distance. This implies that there cannot exist a polynomial upper bound depending only on n and d similar to the Johnson bound in Hamming metric. All three rankmetric bounds reveal significant differences to bounds for codes in Hamming metric.
Hybrid Noncoherent Network Coding
"... We describe a novel extension of subspace codes for noncoherent networks, suitable for use when the network is viewed as a communication system that introduces both dimension and symbol errors. We show that when symbol erasures occur in a significantly large number of different basis vectors transm ..."
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Cited by 2 (0 self)
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We describe a novel extension of subspace codes for noncoherent networks, suitable for use when the network is viewed as a communication system that introduces both dimension and symbol errors. We show that when symbol erasures occur in a significantly large number of different basis vectors transmitted through the network and when the mincut of the networks is much smaller then the length of the transmitted codewords, the new family of codes outperforms their subspace code counterparts. For the proposed coding scheme, termed hybrid network coding, we derive two upper bounds on the size of the codes. These bounds represent a variation of the Singleton and of the spherepacking bound. We show that a simple concatenated scheme that represents a combination of subspace codes and ReedSolomon codes is asymptotically optimal with respect to the Singleton bound. Finally, we describe two efficient decoding algorithms for concatenated subspace codes that in certain cases have smaller complexity than their subspace decoder counterparts. 1
Subspace Codes based on Graph Matchings, Ferrers Diagrams and Pending Blocks
, 2014
"... This paper provides new constructions and lower bounds for subspace codes, using Ferrers diagram rankmetric codes from matchings of the complete graph and pending blocks. We present different constructions for constant dimension codes with minimum injection distance 2 or k − 1, where k is the const ..."
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Cited by 2 (0 self)
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This paper provides new constructions and lower bounds for subspace codes, using Ferrers diagram rankmetric codes from matchings of the complete graph and pending blocks. We present different constructions for constant dimension codes with minimum injection distance 2 or k − 1, where k is the constant dimension. Furthermore, we present a construction of new codes from old codes for any minimum distance. Then we construct nonconstant dimension codes from these codes. The examples of codes obtained by these constructions are the largest known codes for the given parameters.
Message encoding for spread and orbit codes
 In Proceedings of the 2014 IEEE International Symposium on Information Theory (ISIT
, 2014
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