Weight distribution of rank-metric codes

In analogy with the Singleton defect for classical codes, we propose a definition of rank defect for rank-metric codes. We characterize codes whose rank defect and dual rank defect are both zero, and prove that the rank distribution of such codes is determined by their parameters. This extends a res...

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Bibliographic Details
Published inDesigns, codes, and cryptography Vol. 86; no. 1; pp. 1 - 16
Main Authors de la Cruz, Javier, Gorla, Elisa, López, Hiram H., Ravagnani, Alberto
Format Journal Article
LanguageEnglish
Published New York Springer US 2018
Springer Nature B.V
Subjects
Circuits
Codes
Coding and Information Theory
Computer Science
Cryptology
Data Structures and Information Theory
Defects
Discrete Mathematics in Computer Science
Information and Communication
MRD, QMRD, and dually-QMRD codes
68P30
Rank-metric codes
Rank distribution
94B60
94C99
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Summary:In analogy with the Singleton defect for classical codes, we propose a definition of rank defect for rank-metric codes. We characterize codes whose rank defect and dual rank defect are both zero, and prove that the rank distribution of such codes is determined by their parameters. This extends a result by Delsarte on the rank distribution of MRD codes. In the general case of codes of positive defect, we show that the rank distribution is determined by the parameters of the code, together with the number of codewords of small rank. Moreover, we prove that if the rank defect of a code and its dual are both one, and the dimension satisfies a divisibility condition, then the number of minimum-rank codewords and dual minimum-rank codewords is the same. Finally, we discuss how our results specialize to F q m -linear rank-metric codes in vector representation.
ISSN:0925-1022
1573-7586
DOI:10.1007/s10623-016-0325-1