101 research outputs found
Generalized Twisted Gabidulin Codes
Let be a set of by matrices over such
that the rank of is at least for all distinct .
Suppose that . If , then
is a maximum rank distance (MRD for short) code. Until 2016,
there were only two known constructions of MRD codes for arbitrary .
One was found by Delsarte (1978) and Gabidulin (1985) independently, and it was
later generalized by Kshevetskiy and Gabidulin (2005). We often call them
(generalized) Gabidulin codes. Another family was recently obtained by Sheekey
(2016), and its elements are called twisted Gabidulin codes. In the same paper,
Sheekey also proposed a generalization of the twisted Gabidulin codes. However
the equivalence problem for it is not considered, whence it is not clear
whether there exist new MRD codes in this generalization. We call the members
of this putative larger family generalized twisted Gabidulin codes. In this
paper, we first compute the Delsarte duals and adjoint codes of them, then we
completely determine the equivalence between different generalized twisted
Gabidulin codes. In particular, it can be proven that, up to equivalence,
generalized Gabidulin codes and twisted Gabidulin codes are both proper subsets
of this family.Comment: One missing case (n=4) has been included in the appendix. Typos are
corrected, Journal of Combinatorial Theory, Series A, 201
Puncturing maximum rank distance codes
We investigate punctured maximum rank distance codes in cyclic models for bilinear forms of finite vector spaces. In each of these models, we consider an infinite family of linear maximum rank distance codes obtained by puncturing generalized twisted Gabidulin codes. We calculate the automorphism group of such codes, and we prove that this family contains many codes which are not equivalent to any generalized Gabidulin code. This solves a problem posed recently by Sheekey (Adv Math Commun 10:475–488, 2016)
A decoding algorithm for Twisted Gabidulin codes
In this work, we modify the decoding algorithm for subspace codes by Koetter
and Kschischang to get a decoding algorithm for (generalized) twisted Gabidulin
codes. The decoding algorithm we present applies to cases where the code is
linear over the base field but not linear over
.Comment: This paper was submitted to ISIT 201
Equivalence and Characterizations of Linear Rank-Metric Codes Based on Invariants
We show that the sequence of dimensions of the linear spaces, generated by a
given rank-metric code together with itself under several applications of a
field automorphism, is an invariant for the whole equivalence class of the
code. The same property is proven for the sequence of dimensions of the
intersections of itself under several applications of a field automorphism.
These invariants give rise to easily computable criteria to check if two codes
are inequivalent. We derive some concrete values and bounds for these dimension
sequences for some known families of rank-metric codes, namely Gabidulin and
(generalized) twisted Gabidulin codes. We then derive conditions on the length
of the codes with respect to the field extension degree, such that codes from
different families cannot be equivalent. Furthermore, we derive upper and lower
bounds on the number of equivalence classes of Gabidulin codes and twisted
Gabidulin codes, improving a result of Schmidt and Zhou for a wider range of
parameters. In the end we use the aforementioned sequences to determine a
characterization result for Gabidulin codes.Comment: 37 pages, 1 figure, 3 tables, extended version of arXiv:1905.1132
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