11 research outputs found
Communication over Finite-Chain-Ring Matrix Channels
Though network coding is traditionally performed over finite fields, recent
work on nested-lattice-based network coding suggests that, by allowing network
coding over certain finite rings, more efficient physical-layer network coding
schemes can be constructed. This paper considers the problem of communication
over a finite-ring matrix channel , where is the channel
input, is the channel output, is random error, and and are
random transfer matrices. Tight capacity results are obtained and simple
polynomial-complexity capacity-achieving coding schemes are provided under the
assumption that is uniform over all full-rank matrices and is uniform
over all rank- matrices, extending the work of Silva, Kschischang and
K\"{o}tter (2010), who handled the case of finite fields. This extension is
based on several new results, which may be of independent interest, that
generalize concepts and methods from matrices over finite fields to matrices
over finite chain rings.Comment: Submitted to IEEE Transactions on Information Theory, April 2013.
Revised version submitted in Feb. 2014. Final version submitted in June 201
The Smith normal form distribution of a random integer matrix
International audienceWe show that the density μ of the Smith normal form (SNF) of a random integer matrix exists and equals a product of densities μps of SNF over Z/psZ with p a prime and s some positive integer. Our approach is to connect the SNF of a matrix with the greatest common divisors (gcds) of certain polynomials of matrix entries, and develop the theory of multi-gcd distribution of polynomial values at a random integer vector. We also derive a formula for μps and determine the density μ for several interesting types of sets
The Smith Normal Form Distribution of A Random Integer Matrix
We show that the density μ of the Smith normal form (SNF) of a random integer matrix exists and equals a product of densities μ p of SNF over ℤ/p s Z with p a prime and s some positive integer. Our approach is to connect the SNF of a matrix with the greatest common divisors (gcds) of certain polynomials of matrix entries and develop the theory of multi-gcd distribution of polynomial values at a random integer vector. We also derive a formula for μps and compute the density μ for several interesting types of sets. As an application, we determine the probability that the cokernel of a random integer square matrix has at most ℓ generators for a positive integer ℓ, and establish its asymptotics as ℓ → ∞, which extends a result of Ekedahl (1991) on the case ℓ = 1
Algebraic Approach to Physical-Layer Network Coding
The problem of designing physical-layer network coding (PNC) schemes via
nested lattices is considered. Building on the compute-and-forward (C&F)
relaying strategy of Nazer and Gastpar, who demonstrated its asymptotic gain
using information-theoretic tools, an algebraic approach is taken to show its
potential in practical, non-asymptotic, settings. A general framework is
developed for studying nested-lattice-based PNC schemes---called lattice
network coding (LNC) schemes for short---by making a direct connection between
C&F and module theory. In particular, a generic LNC scheme is presented that
makes no assumptions on the underlying nested lattice code. C&F is
re-interpreted in this framework, and several generalized constructions of LNC
schemes are given. The generic LNC scheme naturally leads to a linear network
coding channel over modules, based on which non-coherent network coding can be
achieved. Next, performance/complexity tradeoffs of LNC schemes are studied,
with a particular focus on hypercube-shaped LNC schemes. The error probability
of this class of LNC schemes is largely determined by the minimum inter-coset
distances of the underlying nested lattice code. Several illustrative
hypercube-shaped LNC schemes are designed based on Construction A and D,
showing that nominal coding gains of 3 to 7.5 dB can be obtained with
reasonable decoding complexity. Finally, the possibility of decoding multiple
linear combinations is considered and related to the shortest independent
vectors problem. A notion of dominant solutions is developed together with a
suitable lattice-reduction-based algorithm.Comment: Submitted to IEEE Transactions on Information Theory, July 21, 2011.
Revised version submitted Sept. 17, 2012. Final version submitted July 3,
201
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