595 research outputs found
Low-Rank Parity-Check Codes over the Ring of Integers Modulo a Prime Power
We define and analyze low-rank parity-check (LRPC) codes over extension rings
of the finite chain ring , where is a prime and is a
positive integer. LRPC codes have originally been proposed by Gaborit et
al.(2013) over finite fields for cryptographic applications. The adaption to
finite rings is inspired by a recent paper by Kamche et al. (2019), which
constructed Gabidulin codes over finite principle ideal rings with applications
to space-time codes and network coding. We give a decoding algorithm based on
simple linear-algebraic operations. Further, we derive an upper bound on the
failure probability of the decoder. The upper bound is valid for errors whose
rank is equal to the free rank
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
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
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