11 research outputs found
Systematic maximum sum rank codes
In the last decade there has been a great interest in extending results for codes equipped with the Hamming metric to analogous results for codes endowed with the rank metric. This work follows this thread of research and studies the characterization of systematic generator matrices (encoders) of codes with maximum rank distance. In the context of Hamming distance these codes are the so-called Maximum Distance Separable (MDS) codes and systematic encoders have been fully investigated. In this paper we investigate the algebraic properties and representation of encoders in systematic form of Maximum Rank Distance (MRD) codes and Maximum Sum Rank Distance (MSRD) codes. We address both block codes and convolutional codes separately and present necessary and sufficient conditions for an encoder in systematic form to generate a code with maximum (sum) rank distance. These characterizations are given in terms of certain matrices that must be superregular in a extension field and that preserve superregularity after some transformations performed over the base field. We conclude the work presenting some examples of Maximum Sum Rank convolutional codes over small fields. For the given parameters the examples obtained are over smaller fields than the examples obtained by other authors.publishe
Gabidulin Codes with Support Constrained Generator Matrices
Gabidulin codes are the first general construction of linear codes that are
maximum rank distant (MRD). They have found applications in linear network
coding, for example, when the transmitter and receiver are oblivious to the
inner workings and topology of the network (the so-called incoherent regime).
The reason is that Gabidulin codes can be used to map information to linear
subspaces, which in the absence of errors cannot be altered by linear
operations, and in the presence of errors can be corrected if the subspace is
perturbed by a small rank. Furthermore, in distributed coding and distributed
systems, one is led to the design of error correcting codes whose generator
matrix must satisfy a given support constraint. In this paper, we give
necessary and sufficient conditions on the support of the generator matrix that
guarantees the existence of Gabidulin codes and general MRD codes. When the
rate of the code is not very high, this is achieved with the same field size
necessary for Gabidulin codes with no support constraint. When these conditions
are not satisfied, we characterize the largest possible rank distance under the
support constraints and show that they can be achieved by subcodes of Gabidulin
codes. The necessary and sufficient conditions are identical to those that
appear for MDS codes which were recently proven by Yildiz et al. and Lovett in
the context of settling the GM-MDS conjecture
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
Linear Codes with Constrained Generator Matrices
Designing good error correcting codes whose generator matrix has a support constraint, i.e., one for which only certain entries of the generator matrix are allowed to be nonzero, has found many recent applications, including in distributed coding and storage, linear network coding, multiple access networks, and weakly secure data exchange. The dual problem, where the parity check matrix has a support constraint, comes up in the design of locally repairable codes. The central problem here is to design codes with the largest possible minimum distance, subject to the given support constraint on the generator matrix. When the distance metric is the Hamming distance, the codes of interest are Reed-Solomon codes, for which case, the problem was formulated as the "GM-MDS conjecture." In the rank metric case, the same problem can be considered for Gabidulin codes. This thesis provides solutions to these problems and discusses the remaining open problems.</p