390 research outputs found
Low-Rank Parity-Check Codes over Galois Rings
Low-rank parity-check (LRPC) are rank-metric codes over finite fields, which
have been proposed by Gaborit et al. (2013) for cryptographic applications.
Inspired by a recent adaption of Gabidulin codes to certain finite rings by
Kamche et al. (2019), we define and study LRPC codes over Galois rings - a wide
class of finite commutative rings. We give a decoding algorithm similar to
Gaborit et al.'s decoder, based on simple linear-algebraic operations. We
derive an upper bound on the failure probability of the decoder, which is
significantly more involved than in the case of finite fields. The bound
depends only on the rank of an error, i.e., is independent of its free rank.
Further, we analyze the complexity of the decoder. We obtain that there is a
class of LRPC codes over a Galois ring that can decode roughly the same number
of errors as a Gabidulin code with the same code parameters, but faster than
the currently best decoder for Gabidulin codes. However, the price that one
needs to pay is a small failure probability, which we can bound from above.Comment: 37 pages, 1 figure, extended version of arXiv:2001.0480
Good Random Matrices over Finite Fields
The random matrix uniformly distributed over the set of all m-by-n matrices
over a finite field plays an important role in many branches of information
theory. In this paper a generalization of this random matrix, called k-good
random matrices, is studied. It is shown that a k-good random m-by-n matrix
with a distribution of minimum support size is uniformly distributed over a
maximum-rank-distance (MRD) code of minimum rank distance min{m,n}-k+1, and
vice versa. Further examples of k-good random matrices are derived from
homogeneous weights on matrix modules. Several applications of k-good random
matrices are given, establishing links with some well-known combinatorial
problems. Finally, the related combinatorial concept of a k-dense set of m-by-n
matrices is studied, identifying such sets as blocking sets with respect to
(m-k)-dimensional flats in a certain m-by-n matrix geometry and determining
their minimum size in special cases.Comment: 25 pages, publishe
Erasure Techniques in MRD codes
This book is organized into six chapters. The first chapter introduces the
basic algebraic structures essential to make this book a self contained one.
Algebraic linear codes and their basic properties are discussed in chapter two.
In chapter three the authors study the basic properties of erasure decoding in
maximum rank distance codes. Some decoding techniques about MRD codes are
described and discussed in chapter four of this book. Rank distance codes with
complementary duals and MRD codes with complementary duals are introduced and
their applications are discussed. Chapter five introduces the notion of integer
rank distance codes. The final chapter introduces some concatenation
techniques.Comment: 162 pages; Published by Zip publishing in 201
- …