Cyclic orbit flag codes

In network coding, a flag code is a set of sequences of nested subspaces of F q n , being F q the finite field with q elements. Flag codes defined as orbits of a cyclic subgroup of the general linear group acting on flags of F q n are called cyclic orbit flag codes . Inspired by the ideas in Gluesin...

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Bibliographic Details
Published in:Designs, codes, and cryptography codes, and cryptography, 2021-10, Vol.89 (10), p.2331-2356
Main Authors: Alonso-González, Clementa, Navarro-Pérez, Miguel Ángel
Format: Article
Language:English
Subjects:
Codes
Coding and Information Theory
Computer Science
Cryptology
Discrete Mathematics in Computer Science
Extreme values
Fields (mathematics)
Flags
Subgroups
Subspaces
Vector spaces
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Summary:In network coding, a flag code is a set of sequences of nested subspaces of F q n , being F q the finite field with q elements. Flag codes defined as orbits of a cyclic subgroup of the general linear group acting on flags of F q n are called cyclic orbit flag codes . Inspired by the ideas in Gluesing-Luerssen et al. (Adv Math Commun 9(2):177–197, 2015), we determine the cardinality of a cyclic orbit flag code and provide bounds for its distance with the help of the largest subfield over which all the subspaces of a flag are vector spaces (the best friend of the flag). Special attention is paid to two specific families of cyclic orbit flag codes attaining the extreme possible values of the distance: Galois cyclic orbit flag codes and optimum distance cyclic orbit flag codes . We study in detail both classes of codes and analyze the parameters of the respective subcodes that still have a cyclic orbital structure.
ISSN:0925-1022
1573-7586
DOI:10.1007/s10623-021-00920-5