Joerg Kliewer

Nested Polar Codes for Wiretap and Relay Channels

We show that polar codes asymptotically achieve the whole capacity-equivocation region for the wiretap channel when the wiretappers channel is degraded with respect to the main channel, and the weak secrecy notion is used. Our coding scheme also achieves the capacity of the physically degraded receiver-orthogonal relay channel. We show simulation results for moderate block length for the binary erasure wiretap channel, comparing polar codes and two edge type LDPC codes.

Memoryless Relay Strategies for Two-Way Relay Channels: Performance Analysis and Optimization

We consider relaying strategies for two-way relay channels, where two terminals transmits simultaneously to each other with the help of relays. A memoryless system is considered, where the signal transmitted by a relay depends only on its last received signal. For binary antipodal signaling, we analyze and optimize the performance of existing amplify and forward (AF) and absolute (abs) decode and forward (ADF) for two- way AWGN relay channels. A new abs-based AF (AAF) scheme is proposed, which has better performance than AF. In low SNR, AAF performs even better than ADF. Furthermore, a novel estimate and forward (EF) strategy is proposed which performs better than ADF. More importantly, we optimize the relay strategy within the class of abs-based strategies via functional analysis, which minimizes the average probability of error over all possible relay functions. The optimized function is shown to be a Lamberts W function parameterized on the noise power and the transmission energy. The optimized function behaves like AAF in low SNR and like ADF in high SNR, resp., where EF behaves like the optimized function over the whole SNR range.

Design of Network Codes for Multiple-User Multiple-Relay Wireless Networks

We investigate the design of network codes for multiple-user multiple-relay (MUMR) wireless networks with slow fading (quasi-static) channels. In these networks, M users have independent information to be transmitted to a common base station (BS) with the help of N relays, where M ≥ 2 and N ≥ 1 are arbitrary integers. We investigate such networks in terms of diversity order to measure asymptotic performance. For networks with orthogonal channels, we show that network codes based on maximum distance separable (MDS) codes can achieve the maximum diversity order of N+1. We further show that the MDS coding construction of network codes is also necessary to obtain full diversity for linear finite field network coding (FFNC). Then, we compare the performance of the FFNC approach with superposition coding (SC) at the relays. The results show that the FFNC based on MDS codes has better performance than SC in both the high rate and the high SNR regime. Further, we discuss networks without direct source-to-BS channels for N ≥ M. We show that the proposed FFNC can obtain the diversity order N-M+1, which is equivalent to achieving the Singleton bound for network error-correction codes. Finally, we study the network with nonorthogonal channels and show our codes can still achieve a diversity order of N+1, which cannot be achieved by a scheme based on SC.

On Secure Network Coding With Nonuniform or Restricted Wiretap Sets

The secrecy capacity of a network, for a given collection of permissible wiretap sets, is the maximum rate of communication such that observing links in any permissible wiretap set reveal no information about the message. This paper considers secure network coding with nonuniform or restricted wiretap sets, for example, networks with unequal link capacities where a wiretapper can wiretap any subset of k links, or networks where only a subset of links can be wiretapped. Existing results show that for the case of uniform wiretap sets (networks with equal capacity links/packets where any k can be wiretapped), the secrecy capacity is given by a cut-set bound if random keys are injected at the source (and decoded at the sink), whether or not the communicating users have information about the choice of wiretap set. In contrast, we show that for the nonuniform case, this secrecy rate is achievable for the case of known but not unknown wiretap set. We give achievable linear optimization-based strategies where random keys are canceled at intermediate nonsink nodes or injected at intermediate nonsource nodes. Finally, we show that determining the secrecy capacity is an NP-hard problem.

Spatially coupled sparse codes on graphs: theory and practice

Since the discovery of turbo codes 20 years ago and the subsequent rediscovery of low-density parity check codes a few years later, the field of channel coding has experienced a number of major advances. Until that time, code designers were usually happy with performance that came within a few decibels of the Shannon Limit, primarily due to implementation complexity constraints, whereas the new coding techniques now allow performance within a small fraction of a decibel of capacity with modest encoding and decoding complexity. Due to these significant improvements, coding standards in applications as varied as wireless mobile transmission, satellite TV, and deep space communication are being updated to incorporate the new techniques. In this article, we review a particularly exciting new class of low-density parity check codes called spatially coupled codes, which promise excellent performance over a broad range of channel conditions and decoded error rate requirements.

Strong coordination with polar codes

In this paper, we design explicit codes for strong coordination in two-node networks. Specifically, we consider a two-node network in which the action imposed by nature is binary and uniform, and the action to coordinate is obtained via a symmetric discrete memoryless channel. By observing that polar codes are useful for channel resolvability over binary symmetric channels, we prove that nested polar codes achieve a subset of the strong coordination capacity region, and therefore provide a constructive and low complexity solution for strong coordination.

Performance Analysis and Design of Two Edge-Type LDPC Codes for the BEC Wiretap Channel

We consider transmission over a wiretap channel where both the main channel and the wiretappers channel are binary erasure channels (BEC). A code construction method is proposed using two edge-type low-density parity-check (LDPC) codes based on the coset encoding scheme. Using a single edge-type LDPC ensemble with a given threshold over the BEC, we give a construction for a two edge-type LDPC ensemble with the same threshold. If the given single edge-type LDPC ensemble has degree two variable nodes, our construction gives rise to degree one variable nodes in the code used over the main channel. This results in zero threshold over the main channel. In order to circumvent this problem, the degree distribution of the two edge-type LDPC ensemble is numerically optimized. We find that the resulting ensembles are able to perform close to the boundary of the rate-equivocation region of the wiretap channel. Further, a method to compute the ensemble average equivocation of two edge-type LDPC ensembles is provided by generalizing a recently published approach to measure the equivocation of single edge-type ensembles for transmission over the BEC in the point-to-point setting. From this analysis, we find that relatively simple constructions give very good secrecy performance.

Multiple-Access Network Information-Flow and Correction Codes

This work considers the multiple-access multicast error-correction scenario over a packetized network with z malicious edge adversaries. The network has min-cut m and packets of length l, and each sink demands all information from the set of sources S. The capacity region is characterized for both a “side-channel” model (where sources and sinks share some random bits that are secret from the adversary) and an “omniscient” adversarial model (where no limitations on the adversarys knowledge are assumed). In the “side-channel” adversarial model, the use of a secret channel allows higher rates to be achieved compared to the “omniscient” adversarial model, and a polynomial-complexity capacity-achieving code is provided. For the “omniscient” adversarial model, two capacity-achieving constructions are given: the first is based on random subspace code design and has complexity exponential in lm, while the second uses a novel multiple-field-extension technique and has O(lm|S|) complexity, which is polynomial in the network size. Our code constructions are “end-to-end” in that all nodes except the sources and sinks are oblivious to the adversaries and may simply implement predesigned linear network codes (random or otherwise). Also, the sources act independently without knowledge of the data from other sources.

Rate rRegions for coherent and noncoherent multisource network error correction

In this paper we derive capacity regions for network error correction with both known and unknown topologies (coherent and non-coherent network coding) under a multiple-source multicast transmission scenario. For the multiple-source non-multicast scenario, given any achievable network code for the error-free case, we construct a code with a reduced rate region for the case with errors.

On secure network coding with uniform wiretap sets

This paper studies secure unicast communication over a network with uniform wiretap sets and shows that, when network nodes can independently generate randomness, determining the secrecy capacity is at least as difficult as the k-unicast network coding problem. In particular, we show that a general k-unicast problem can be reduced to the problem of finding the secrecy capacity of a corresponding single unicast network with uniform link capacities and any one wiretap link. We propose a low-complexity linear optimization-based achievable strategy involving global random keys that can be generated anywhere in the network, and an efficient greedy algorithm that further improves achievable rate by exploiting local random keys.