Optimum Linear Codes With Support-Constrained Generator Matrices Over Small Fields

Hikmet Yildiz; Babak Hassibi

doi:10.1109/tit.2019.2932663

Optimum Linear Codes With Support-Constrained Generator Matrices Over Small Fields

Hikmet Yildiz, Babak Hassibi

Research output: Contribution to journal › Article › peer-review

12 Scopus citations

Abstract

We consider the problem of designing optimal linear codes (in terms of having the largest minimum distance) subject to a support constraint on the generator matrix. We show that the largest minimum distance can be achieved by a subcode of a Reed-Solomon code of small field size and with the same minimum distance. In particular, if the code has length n , and maximum minimum distance d (over all generator matrices with the given support), then an optimal code exists for any field size q≥ 2n-d. As a by-product of this result, we settle the GM-MDS conjecture in the affirmative.

Original language	English (US)
Pages (from-to)	7868-7875
Number of pages	8
Journal	IEEE Transactions on Information Theory
Volume	65
Issue number	12
DOIs	https://doi.org/10.1109/tit.2019.2932663
State	Published - Dec 2019
Externally published	Yes

Bibliographical note

KAUST Repository Item: Exported on 2021-04-10
Acknowledgements: This work was supported in part by the National Science Foundation under Grant CNS-0932428, Grant CCF-1018927, Grant CCF1423663, 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 publication acknowledges KAUST support, but has no KAUST affiliated authors.

ASJC Scopus subject areas

Library and Information Sciences
Information Systems
Computer Science Applications

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10.1109/tit.2019.2932663

Cite this

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title = "Optimum Linear Codes With Support-Constrained Generator Matrices Over Small Fields",

abstract = "We consider the problem of designing optimal linear codes (in terms of having the largest minimum distance) subject to a support constraint on the generator matrix. We show that the largest minimum distance can be achieved by a subcode of a Reed-Solomon code of small field size and with the same minimum distance. In particular, if the code has length n , and maximum minimum distance d (over all generator matrices with the given support), then an optimal code exists for any field size q≥ 2n-d. As a by-product of this result, we settle the GM-MDS conjecture in the affirmative.",

author = "Hikmet Yildiz and Babak Hassibi",

note = "KAUST Repository Item: Exported on 2021-04-10 Acknowledgements: This work was supported in part by the National Science Foundation under Grant CNS-0932428, Grant CCF-1018927, Grant CCF1423663, 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 publication acknowledges KAUST support, but has no KAUST affiliated authors.",

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N2 - We consider the problem of designing optimal linear codes (in terms of having the largest minimum distance) subject to a support constraint on the generator matrix. We show that the largest minimum distance can be achieved by a subcode of a Reed-Solomon code of small field size and with the same minimum distance. In particular, if the code has length n , and maximum minimum distance d (over all generator matrices with the given support), then an optimal code exists for any field size q≥ 2n-d. As a by-product of this result, we settle the GM-MDS conjecture in the affirmative.

AB - We consider the problem of designing optimal linear codes (in terms of having the largest minimum distance) subject to a support constraint on the generator matrix. We show that the largest minimum distance can be achieved by a subcode of a Reed-Solomon code of small field size and with the same minimum distance. In particular, if the code has length n , and maximum minimum distance d (over all generator matrices with the given support), then an optimal code exists for any field size q≥ 2n-d. As a by-product of this result, we settle the GM-MDS conjecture in the affirmative.

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Optimum Linear Codes With Support-Constrained Generator Matrices Over Small Fields

Abstract

Bibliographical note

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