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Gabidulin Codes with Support Constrained Generator Matrices

Yildiz, Hikmet and Hassibi, Babak (2020) Gabidulin Codes with Support Constrained Generator Matrices. IEEE Transactions on Information Theory, 66 (6). pp. 3638-3649. ISSN 0018-9448. doi:10.1109/tit.2019.2955106.

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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.

Item Type:Article
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URLURL TypeDescription Paper
Yildiz, Hikmet0000-0002-0891-3352
Additional Information:© 2019 IEEE. Manuscript received April 13, 2019; revised November 16, 2019; accepted November 17, 2019. Date of publication November 22, 2019; date of current version May 20, 2020. This work was supported in part by the National Science Foundation under Grant CNS-0932428, Grant CCF-1018927, Grant CCF-1423663, and Grant CCF-1409204; in part by Qualcomm Inc.; in part by the NASAs Jet Propulsion Laboratory through the President and Directors Fund; and in part by the King Abdullah University of Science and Technology. This work was presented in part at the 2019 Information Theory Workshop.
Funding AgencyGrant Number
JPL President and Director's FundUNSPECIFIED
King Abdullah University of Science and Technology (KAUST)UNSPECIFIED
Subject Keywords:Gabidulin codes, rank distance, network coding, distributed systems
Issue or Number:6
Record Number:CaltechAUTHORS:20190401-161416001
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Official Citation:H. Yildiz and B. Hassibi, "Gabidulin Codes With Support Constrained Generator Matrices," in IEEE Transactions on Information Theory, vol. 66, no. 6, pp. 3638-3649, June 2020, doi: 10.1109/TIT.2019.2955106.
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:94336
Deposited By: Tony Diaz
Deposited On:01 Apr 2019 23:39
Last Modified:16 Nov 2021 17:04

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