37 research outputs found
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
Twisted Reed-Solomon Codes
We present a new general construction of MDS codes over a finite field
. We describe two explicit subclasses which contain new MDS codes
of length at least for all values of . Moreover, we show that
most of the new codes are not equivalent to a Reed-Solomon code.Comment: 5 pages, accepted at IEEE International Symposium on Information
Theory 201
MRD codes with maximum idealizers
Left and right idealizers are important invariants of linear rank-distance
codes. In the case of maximum rank-distance (MRD for short) codes in
the idealizers have been proved to be isomorphic to
finite fields of size at most . Up to now, the only known MRD codes with
maximum left and right idealizers are generalized Gabidulin codes, which were
first constructed in 1978 by Delsarte and later generalized by Kshevetskiy and
Gabidulin in 2005. In this paper we classify MRD codes in
for with maximum left and right idealizers
and connect them to Moore-type matrices. Apart from generalized Gabidulin
codes, it turns out that there is a further family of rank-distance codes
providing MRD ones with maximum idealizers for , odd and for ,
. These codes are not equivalent to any previously known MRD
code. Moreover, we show that this family of rank-distance codes does not
provide any further examples for .Comment: Reviewers' comments implemented, we changed the titl