1,113 research outputs found
Diversity analysis, code design, and tight error rate lower bound for binary joint network-channel coding
Joint network-channel codes (JNCC) can improve the performance of communication in wireless networks, by combining, at the physical layer, the channel codes and the network code as an overall error-correcting code. JNCC is increasingly proposed as an alternative to a standard layered construction, such as the OSI-model. The main performance metrics for JNCCs are scalability to larger networks and error rate. The diversity order is one of the most important parameters determining the error rate. The literature on JNCC is growing, but a rigorous diversity analysis is lacking, mainly because of the many degrees of freedom in wireless networks, which makes it very hard to prove general statements on the diversity order. In this article, we consider a network with slowly varying fading point-to-point links, where all sources also act as relay and additional non-source relays may be present. We propose a general structure for JNCCs to be applied in such network. In the relay phase, each relay transmits a linear transform of a set of source codewords. Our main contributions are the proposition of an upper and lower bound on the diversity order, a scalable code design and a new lower bound on the word error rate to assess the performance of the network code. The lower bound on the diversity order is only valid for JNCCs where the relays transform only two source codewords. We then validate this analysis with an example which compares the JNCC performance to that of a standard layered construction. Our numerical results suggest that as networks grow, it is difficult to perform significantly better than a standard layered construction, both on a fundamental level, expressed by the outage probability, as on a practical level, expressed by the word error rate
Reed-Muller codes for random erasures and errors
This paper studies the parameters for which Reed-Muller (RM) codes over
can correct random erasures and random errors with high probability,
and in particular when can they achieve capacity for these two classical
channels. Necessarily, the paper also studies properties of evaluations of
multi-variate polynomials on random sets of inputs.
For erasures, we prove that RM codes achieve capacity both for very high rate
and very low rate regimes. For errors, we prove that RM codes achieve capacity
for very low rate regimes, and for very high rates, we show that they can
uniquely decode at about square root of the number of errors at capacity.
The proofs of these four results are based on different techniques, which we
find interesting in their own right. In particular, we study the following
questions about , the matrix whose rows are truth tables of all
monomials of degree in variables. What is the most (resp. least)
number of random columns in that define a submatrix having full column
rank (resp. full row rank) with high probability? We obtain tight bounds for
very small (resp. very large) degrees , which we use to show that RM codes
achieve capacity for erasures in these regimes.
Our decoding from random errors follows from the following novel reduction.
For every linear code of sufficiently high rate we construct a new code
, also of very high rate, such that for every subset of coordinates, if
can recover from erasures in , then can recover from errors in .
Specializing this to RM codes and using our results for erasures imply our
result on unique decoding of RM codes at high rate.
Finally, two of our capacity achieving results require tight bounds on the
weight distribution of RM codes. We obtain such bounds extending the recent
\cite{KLP} bounds from constant degree to linear degree polynomials
Performance Analysis of Random Linear Network Coding in Two-Source Single-Relay Networks
This paper considers the multiple-access relay channel in a setting where two
source nodes transmit packets to a destination node, both directly and via a
relay node, over packet erasure channels. Intra-session network coding is used
at the source nodes and inter-session network coding is employed at the relay
node to combine the recovered source packets of both source nodes. In this
work, we investigate the performance of the network-coded system in terms of
the probability that the destination node will successfully recover the source
packets of the two source nodes. We build our analysis on fundamental
probability expressions for random matrices over finite fields and we derive
upper bounds on the system performance for the case of systematic and
non-systematic network coding. Simulation results show that the upper bounds
are very tight and accurately predict the decoding probability at the
destination node. Our analysis also exposes the clear benefits of systematic
network coding at the source nodes compared to non-systematic transmission.Comment: Proc. ICC 2015, Workshop on Cooperative and Cognitive Mobile Networks
(CoCoNet), to appea
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