Gabidulin Codes With Support Constrained Generator Matrices

Gabidulin codes are the first general construction of linear codes that are maximum rank distant (MRD). They have found applications in linear network coding, for example, when the transmitter and receiver are oblivious to the inner workings and topology of the network (the so-called incoherent regi...

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Bibliographic Details
Published in:IEEE transactions on information theory 2020-06, Vol.66 (6), p.3638-3649
Main Authors: Yildiz, Hikmet, Hassibi, Babak
Format: Article
Language:English
Subjects:
Codes
Coding
Computer networks
Constraints
distributed systems
Error correcting codes
Error correction
Gabidulin codes
Generators
Hamming distance
Linear codes
Mathematical analysis
Matrix methods
Measurement
Network coding
rank distance
Receivers
Subspaces
Topology
Transmitters
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Summary:Gabidulin codes are the first general construction of linear codes that are maximum rank distant (MRD). They have found applications in linear network coding, for example, when the transmitter and receiver are oblivious to the inner workings and topology of the network (the so-called incoherent regime). The reason is that Gabidulin codes can be used to map information to linear subspaces, which in the absence of errors cannot be altered by linear operations, and in the presence of errors can be corrected if the subspace is perturbed by a small rank. Furthermore, in distributed coding and distributed systems, one is led to the design of error correcting codes whose generator matrix must satisfy a given support constraint. In this paper, we give necessary and sufficient conditions on the support of the generator matrix that guarantees the existence of Gabidulin codes and general MRD codes. When the rate of the code is not very high, this is achieved with the same field size necessary for Gabidulin codes with no support constraint. When these conditions are not satisfied, we characterize the largest possible rank distance under the support constraints and show that they can be achieved by subcodes of Gabidulin codes. The necessary and sufficient conditions are identical to those that appear for MDS codes which were recently proven by Yildiz et al. and Lovett in the context of settling the GM-MDS conjecture.
ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2019.2955106