71,908 research outputs found
Further Generalisations of Twisted Gabidulin Codes
We present a new family of maximum rank distance (MRD) codes. The new class
contains codes that are neither equivalent to a generalised Gabidulin nor to a
twisted Gabidulin code, the only two known general constructions of linear MRD
codes.Comment: 10 pages, accepted at the International Workshop on Coding and
Cryptography (WCC) 201
A new family of maximum scattered linear sets in PG(1,q^6)
We generalize the example of linear set presented by the last two authors in "Vertex properties of maximum scattered linear sets of PG(1; qn)" (2019) to a more general family, proving that such linear sets are maximum scattered when q is odd and, apart from a special case, they are new. This solves an open problem posed in "Vertex properties of maximum scattered linear sets of PG(1; qn)" (2019). As a consequence of Sheekey's results in "A new family of linear maximum rank distance codes" (2016), this family yields to new MRD-codes with parameters (6; 6; q; 5)
A new family of maximum scattered linear sets in
We generalize the example of linear set presented by the last two authors in
"Vertex properties of maximum scattered linear sets of "
(2019) to a more general family, proving that such linear sets are maximum
scattered when is odd and, apart from a special case, they are are new.
This solves an open problem posed in "Vertex properties of maximum scattered
linear sets of " (2019). As a consequence of Sheekey's
results in "A new family of linear maximum rank distance codes" (2016), this
family yields to new MRD-codes with parameters
MRD Rank Metric Convolutional Codes
So far, in the area of Random Linear Network Coding, attention has been given to the so-called one-shot network coding, meaning that the network is used just once to propagate the information. In contrast, one can use the network more than once to spread redundancy over different shots. In
this paper, we propose rank metric convolutional codes for this purpose. The framework we present is slightly more general than the one which can be found in the literature. We introduce a rank distance, which is suitable for convolutional codes, and derive a new Singleton-like upper bound. Codes achieving this bound are called Maximum Rank Distance (MRD) convolutional codes. Finally, we prove that this bound is optimal by showing a concrete construction of a family of MRD convolutional codes
Codes and Designs Related to Lifted MRD Codes
Lifted maximum rank distance (MRD) codes, which are constant dimension codes,
are considered. It is shown that a lifted MRD code can be represented in such a
way that it forms a block design known as a transversal design. A slightly
different representation of this design makes it similar to a analog of a
transversal design. The structure of these designs is used to obtain upper
bounds on the sizes of constant dimension codes which contain a lifted MRD
code. Codes which attain these bounds are constructed. These codes are the
largest known codes for the given parameters. These transversal designs can be
also used to derive a new family of linear codes in the Hamming space. Bounds
on the minimum distance and the dimension of such codes are given.Comment: Submitted to IEEE Transactions on Information Theory. The material in
this paper was presented in part in the 2011 IEEE International Symposium on
Information Theory, Saint Petersburg, Russia, August 201
Skew and linearized Reed-Solomon codes and maximum sum rank distance codes over any division ring
Reed-Solomon codes and Gabidulin codes have maximum Hamming distance and
maximum rank distance, respectively. A general construction using skew
polynomials, called skew Reed-Solomon codes, has already been introduced in the
literature. In this work, we introduce a linearized version of such codes,
called linearized Reed-Solomon codes. We prove that they have maximum sum-rank
distance. Such distance is of interest in multishot network coding or in
singleshot multi-network coding. To prove our result, we introduce new metrics
defined by skew polynomials, which we call skew metrics, we prove that skew
Reed-Solomon codes have maximum skew distance, and then we translate this
scenario to linearized Reed-Solomon codes and the sum-rank metric. The theories
of Reed-Solomon codes and Gabidulin codes are particular cases of our theory,
and the sum-rank metric extends both the Hamming and rank metrics. We develop
our theory over any division ring (commutative or non-commutative field). We
also consider non-zero derivations, which give new maximum rank distance codes
over infinite fields not considered before
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