We study the performance of polarizing codes over a degraded symmetric wiretap channel under a total variation distance (TVD) secrecy constraint. We show that the leakage can be bounded by the sum of the TVDs of the bit-channels corresponding to the confidential and frozen bits. In the asymptotic regime, this gives a new criterion to design wiretap codes with vanishing TVD leakage. In finite blocklength, it allows us to compute lower bounds for the secrecy rate of different families of polarizing wiretap codes over a binary erasure wiretap channel.

Automorphism Ensemble (AE) decoding has recently drawn attention as a possible alternative to list decoding of polar codes. In this letter, we investigate the distribution of Partially-Symmetric Reed-Muller (PS-RM) codes, a family of polar codes yielding good performances under AE decoding. We prove the existence of these codes for almost all code dimensions for code lengths N ≤ 256. Moreover, we analyze the absorption group of this family of codes under SC decoding, proving that valuable permutations in AE decoding always exist. Finally, we experimentally show that PS-RM codes can outperform state-of-the-art polar-code-construction algorithms in terms of error-correction performance for short code lengths, while reducing decoding latency.

In this paper, we propose an analysis of the automorphism group of polar codes, with the aim of designing codes tailored forautomorphism ensemble(AE) decoding. Using a novel description of polar codes as monomial codes through negative monomials, we prove the equivalence between the notion ofdecreasing monomial codesand the universal partial order (UPO) framework for polar codes; this property is widely believed to hold true but a formal proof was missing. We further provide a rigorous mathematical connection between code word permutations and affine transformations, an important link to understand the considered automorphisms. Based on this mathematical formalisms, we analyze the algebraic properties of theaffine automorphisms groupof polar codes, providing a novel description of its structure. We classify automorphisms such that all automorphisms in the same class lead to the same result under permutation decoding, which gives rise to the concept ofredundantautomorphisms. Mathematically this is achieved by introducing equivalence classes of affine automorphisms under AE-based decoding. For practical application, we provide an algorithm to compute representatives for the equivalence classes, such that one automorphism from each equivalence class can be selected for use in AE decoding. A numerical analysis of the error correction performance of AE decoding of polar codes, based on equivalence classes, concludes the paper.

This paper proposes new polar code design principles for the low-latency automorphism ensemble (AE) decoding. Our proposal permits to design a polar code with the desired automorphism group (if possible) while assuring the decreasing monomial property. Moreover, we prove that some automorphisms are redundant under AE decoding, and we propose a new automorphisms classification based on equivalence classes. Finally, we propose an automorphism selection heuristic based on drawing only one element of each class; we show that this method enhances the block error rate (BLER) performance of short polar codes even with a limited number of automorphisms.

We propose a novel coupling technique for polar codes via a special kernel that enables efficient sliding window decoding. This feature allows to reduce the memory requirement of the decoder, an important possibility in wireless communication downlink scenarios. Our approach is based on the design of an ad-hoc kernel to be inserted in a multi-kernel polar code framework. Simulation results show that the proposed sliding window polar codes outperform polar codes transmitted in independent blocks at a negligible additional decoding overhead.

In this paper we deal with polar code automorphisms that are beneficial under low-latency automorphism ensemble (AE) decoding, and we propose polar code designs that have such automorphisms. Successive-cancellation (SC) decoding and thus SC-based AE decoding are invariant with respect to the only known polar code automorphisms, namely those of the lower-triangular affine (LTA) group. To overcome this problem, we provide methods to determine whether a given polar code has non-LTA automorphisms and to identify such automorphisms. Building on this, we design specific polar codes that admit automorphisms in the upper-diagonal linear (UTL) group, and thus render SC-based AE decoding effective. Demonstrated by examples, these new polar codes under AE decoding outperform conventional polar codes under SC list decoding in terms of error rate, while keeping the latency comparable to SC decoding. Moreover, state-of-the-art BP-based permutation decoding for polar codes is beaten by BP-based AE thanks to this design.

In this paper we propose an enhanced soft cancellation (SCAN) decoder for polar codes based on decoding stages permutation. The proposed soft cancellation list (SCANL) decoder runs L independent SCAN decoders, each one relying on a different permuted factor graph. The estimated bits are selected among the L candidates through a dedicated metric provided by the decoders. Furthermore, we introduce an early-termination scheme reducing decoding latency without affecting error correction performance. We investigate the error-correction performance of the proposed scheme under various combinations of number of iterations used, permutation set and early-termination condition. Simulation results show that the proposed SCANL provides similar results when compared with belief propagation list, while having a smaller complexity. Moreover, for large list sizes, SCANL outperforms non-CRC aided successive cancellation list decoding.

The 5thgeneration wireless systems (5G) standardization process of the 3rdgeneration partnership project (3GPP) chose polar codes as a channel coding scheme for the control channel. In case of downlink control information, polar codes are concatenated with distributed distributed cyclic redundancy check (CRC). Whereas CRC bits allow to improve the performance of successive cancellation list (SCL) decoders by improving distance properties, distributed CRC bits allow for path pruning and decoding early-termination. In this paper, we show how to take advantage of the distributed CRC to improve SCL decoding, analyzing various schemes having different early-termination and error correction properties. Simulation results compare the proposed decoding schemes, showing different tradeoffs between error-correction performance and early-termination with different decoder parameters.

In this paper, we propose a new polar code construction by employing kernels of different sizes in the Kronecker product of the transformation matrix, thus generalizing the original construction by Arikan. These multi-kernel polar codes allow for more flexibility in terms of the code length and for various new design principles. Next to the common reliability design, we provide a design to maximize the minimal distance and a hybrid design combining reliability and distance properties. Numerical results demonstrate the advantage of multi-kernel polar codes under the new design principles compared to punctured and shortened Arikan polar codes.

In this paper we propose a novel frozen set design for polar codes and multi-kernel polar codes. We improve the existing hybrid distance-reliability design by minimizing the upper bound of the overall system error probability instead of minimizing its lower bound as previously proposed. This allows to better trade reliabilities of the input bits against distance properties of the code. We describe the new design approach, propose a greedy algorithm to limit the complexity of the code construction process, and evaluate its performance through numerical examples. In both MK polar codes and conventional polar codes, a substantial performance improvement is observed, matching the performance of CRC-aided polar codes under SCL without the need for a CRC.

The successive cancellation flip (SC-Flip) decoding algorithm arose as a valid low-complexity decoding algorithm for polar codes, however its decoding capabilities are still far away from list based decoders. In this paper, we propose an improved SC-Flip multiple error decoding framework based on error dependency, and specialize it for the two-error correction case. We propose to generate different lists of second error locations based on the index of the expected first errors. The inherent flexibility of this approach allows it to be modified for a desired trade-off between performance and complexity. Two second errors list construction approaches are presented, and are shown to yield gains over the original SC-Flip decoder at the same decoding complexity.

Product codes are widespread in optical communications, thanks to their high throughput and good error-correction performance. Systematic polar codes have been recently considered as component codes for product codes. In this paper, we present a novel construction for product polar codes based on non-systematic polar codes. We prove that the resulting product code is actually a polar code, having a frozen set that is dependent on the frozen sets of the component polar codes. We propose a low-complexity decoding algorithm exploiting the dual nature of the constructed code. Performance analysis and simulations show high decoding speed, that allows to construct long codes while maintaining low decoding latency. The resulting high throughput and good error-correction performance are appealing for optical communication systems and other systems where high throughput and low latency are required.

The majority of modern communication systems adopt quadrature amplitude modulation (QAM) constellations as transmission schemes. Due to their square structure, however, QAM do not provide satisfying protection to phase noise effects, as the number of constellation points grows, increasing at the same time their peak-to-average-power ratio. This requires an expensive power amplifier and an oscillator at the transmitter to guarantee low distortion, complicating the adoption of dense transmission schemes in practical high-data rate systems. In this paper, we construct a coded modulation scheme based on regular amplitude and phase shift keying modulations. We propose a novel multilevel coding labeling for the constellation points separating the amplitude and phase domains. We provide a novel multistage decoding scheme allowing for a low-complexity log-likelihood ratio calculation for soft-input decoding of component codes, along with a suitable rate design. Finally, we compare the proposed scheme with the state-of-the-art QAM constellations and optimize the constellations in the presence of phase noise.

Research on polar codes has been constantly gaining attention over the last decade, by academia and industry alike, thanks to their capacity-achieving error-correction performance and low-complexity decoding algorithms. Recently, they have been selected as one of the coding schemes in the 5th generation wireless standard (5G). Over the years various polar code decoding algorithms, like SC-list (SCL), have been proposed to improve the mediocre performance of the successive cancellation (SC) decoding algorithm for finite code lengths; however, like SC, they suffer from long decoding latency. Fast decoding of polar codes tries to overcome this problem by identifying particular subcodes in the polar code and decoding them with efficient decoders. In this work, we introduce a generalized approach to fast decoding of polar codes to further reduce SC-based decoding latency. We propose three multi-node polar code subcodes whose identification patterns include most of the existing subcodes, extending them to SCL decoding, and allow to apply fast decoding to larger subsets of bits. Without any error-correction performance degradation, the proposed technique shows up to 23.6% and 29.2% decoding latency gain with respect to fast SC and SCL decoding algorithms, respectively, and up to 63.6% and 49.8% if a performance loss is accepted, whose amount depends on code and decoding algorithm parameters, along with the desired speedup.

In this paper, we propose a construction for multi-kernel polar codes based on the maximization of the minimum distance. Compared to the original construction based on density evolution, our new design shows particular advantages for short code lengths, where the polarization effect has less impact on the performance than the distances of the code. We introduce and compute the minimum-distance profile and provide a simple greedy algorithm for the code design. Compared to state-of-the-art punctured or shortened Arikan polar codes, multi-kernel polar codes with our new design show significantly improved error-rate performance.

In this paper we propose a novel online caching method to perform the update phase in a distributed caching system, with a natural application to heterogeneous scenarios with cache-equipped small-cell base stations. We investigate the performance of our scheme, showing in particular that it results in a significant reduction of backhaul load, to the point of converging to the performance of the optimal offline placement scheme. While the optimal scheme is obtained at a high complexity cost, the performance of our solution is achieved in a low-complexity decentralized manner. Numerical simulations confirm that the novel online scheme adapts efficiently to varying networks parameters such as the file popularities, without the need of the usually time-consuming learning phase, which makes the proposal adapted for low-latency caching applications.

In this paper, we investigate a novel family of polar codes based on multi-kernel constructions, proving that this construction actually polarizes. To this end, we derive a new and more general proof of polarization, which gives sufficient conditions for kernels to polarize. Finally, we derive the convergence rate of the multi-kernel construction and relate it to the convergence rate of each of the constituent kernels.

In this paper we investigate the problem of optimal cache placement under secrecy constraints in heterogeneous networks, where small-cell base stations are equipped with caches to reduce the overall backhaul load. For two models for eavesdropping attacks, we formally derive the necessary conditions for secrecy and we derive the corresponding achievable backhaul rate. In particular we formulate the optimal caching schemes with secrecy constraints as a convex optimization problem. We then thoroughly investigate the backhaul rate performance of the heterogeneous network with secrecy constraints using numerical simulations. We compare the system performance with and without secrecy constraints and we analyze the influence of the system parameters, such as the file popularity, size of the library files and the capabilities of the small-cell base stations, on the overall performance of our optimal caching strategy. Our results highlight the considerable impact of the secrecy requirements on the overall caching performance of the network.

In this paper, we study the problem of content placement for caching at the wireless edge with the goal to maximize the energy efficiency (EE) of heterogeneous wireless networks. In particular, we consider the minimization of two fundamental metrics: the expected backhaul rate and the energy consumption. We derive both metrics in closed-form expressions, and we solve the minimization problem as a convex optimization for each, highlighting the existence of a tradeoff between the two metrics. Further, we show the advantage of encoding the data using maximum-distance separable (MDS) codes over the alternative concept of file fragmentation, with respect to both backhaul rate and energy consumption. Then, we thoroughly study the performance of the optimal MDS-encoded caching scheme in terms of overall energy consumption for an important heterogeneous network scenario. We compare our optimal strategy to several other sub-optimal caching strategies, including the caching scheme, minimizing the backhaul rate, and we analyze the effects of the system parameters on the overall performance. Our analysis can be generalized to any network topology and to any small-cell base station capability. Our results show that the optimal placement of MDS-encoded content in caches at the wireless edge increases significantly the overall EE of the heterogeneous network. This demonstrates the importance of the edge caching strategy for energy-efficient network designs.

In this paper, the security of the compressed sensing (CS) framework as a form of data confidentiality is analyzed. Two important properties of one-time random linear measurements acquired using a Gaussian independent identically distributed matrix are outlined: 1) the measurements reveal only the energy of the sensed signal and 2) only the energy of the measurements leaks information about the signal. An important consequence of the above facts is that CS provides information theoretic secrecy in a particular setting. Namely, a simple strategy based on the normalization of the Gaussian measurements achieves, at least in theory, perfect secrecy, enabling the use of CS as an additional security layer in privacy preserving applications. In the generic setting in which CS does not provide information theoretic secrecy, two alternative security notions linked to the difficulty of estimating the energy of the signal and distinguishing equal-energy signals are introduced. Useful bounds on the mean square error of any possible estimator and the probability of error of any possible detector are provided and compared with the simulations. The results indicate that CS is in general not secure according to cryptographic standards, but may provide a useful built-in data obfuscation layer.

In this paper we investigate the problem of optimal MDS-encoded cache placement at the wireless edge to minimize the backhaul rate in heterogeneous networks. We derive the backhaul rate performance of any caching scheme based on file splitting and MDS encoding and we formulate the optimal caching scheme as a convex optimization problem. We then thoroughly investigate the performance of this optimal scheme for an important heterogeneous network scenario. We compare it to several other caching strategies and we analyze the influence of the system parameters, such as the popularity and size of the library files and the capabilities of the small-cell base stations, on the overall performance of our optimal caching strategy. Our results show that the careful placement of MDS-encoded content in caches at the wireless edge leads to a significant decrease of the load of the network backhaul and hence to a considerable performance enhancement of the network.

In this paper, we propose a mathematical model to estimate the sparsity degree k of exactly k-sparse signals acquired through Compressed Sensing (CS). Our method does not need to recover the signal to estimate its sparsity, and is based on the use of sparse sensing matrices. We exploit this model to propose a CS acquisition system where the number of measurements is calculated on-the-fly depending on the estimated signal sparsity. Experimental results on block-based CS acquisition of black and white images show that the proposed adaptive technique outperforms classical CS acquisition methods where the number of measurements is set a priori.

In this paper, we analyze the security of compressed sensing (CS) as a cryptosystem. We demonstrate that random linear measurements acquired using a Gaussian i.i.d. matrix reveal only the energy of the sensed signal, and that only the energy of the measurements leaks information about the signal. We provide useful bounds for assessing the information leakage about the energy, linking those bounds to the minimum mean square error achievable by practical estimators. Moreover, we propose a simple strategy based on the normalization of the measurements which achieves, at least in theory, perfect secrecy, enabling the use of CS-based encryption in practical cryptosystems.

In this paper we present F2OMP, a recovery algorithm for Compressed Sensing over finite fields. Classical recovery algorithms do not exploit the fact that a signal may belong to a finite alphabet, while we show that this information can lead to more efficient reconstruction algorithms. As an application, we use the proposed algorithm to recover sparse grayscale images, showing that performing CS operation over a finite field can outperform classical recovery algorithms from visual quality, memory occupation and complexity point of view.

A key problem in network coding (NC) lies in the complexity and energy consumption associated with the packet decoding processes, which hinder its application in mobile environments. Controlling and hence limiting such factors has always been an important but elusive research goal, since the packet degree distribution, which is the main factor driving the complexity, is altered in a non-deterministic way by the random recombinations at the network nodes. In this paper we tackle this problem with a new approach and propose Band Codes (BC), a novel class of network codes specifically designed to preserve the packet degree distribution during packet encoding, recombination and decoding. BC are random codes over GF(2) that exhibit low decoding complexity, feature limited and controlled degree distribution by construction, and hence allow to effectively apply NC even in energy-constrained scenarios. In particular, in this paper we motivate and describe our new design and provide a thorough analysis of its performance. We provide numerical simulations of the BC performance in order to validate the analysis and assess the overhead of BC with respect to a conventional random NC scheme. Moreover, experiment in a real-world application, namely peer-to-peer mobile media streaming using a random-push protocol, show that BC reduce the decoding complexity by a factor of two with negligible increase of the encoding overhead, paving the way for the application of NC to power-constrained devices.

Peer-to-peer (P2P) resource location techniques in grid systems have been recently investigated to obtain scalability, reliability, efficiency, fault-tolerance, security, and robustness. Query resolution for locating resources and update information on their own resource status in these systems can be abstracted as the problem of allowing one peer to obtain a local view of global information defined on all peers of a P2P unstructured network. In this paper, the system is represented as a set of nodes connected to form a P2P network where each node holds a piece of information that is required to be communicated to all the participants. Moreover, we assume that the information can dynamically change and that each peer periodically requires to access the values of the data of all other peers. A novel approach based on a continuous flow of control packets exchanged among the nodes using the random walk principle and rateless coding is proposed. An innovative rateless decoding mechanism that is able to cope with asynchronous information updates is also proposed. The performance of the proposed system is evaluated both analytically and experimentally by simulation. The analytical results show that the proposed strategy guarantees quick diffusion of the information and scales well to large networks. Simulations show that the technique is effective also in presence of network and information dynamics.

In this work, we propose a feedback scheme to drive the packet recombination process at the network nodes in a collaborative Network Coding (NC) scenario. Our scheme addresses the issue of determining which symbols are more helpful at the receivers to recover the message and how to accordingly recombine the received packets at the intermediate nodes where the original symbols are not available. We experimentally demonstrate that our scheme increases the coding efficiency and reduces the computational complexity at the decoder in a video communication scenario without using explicit feedback messages.

We present Band Codes (BC), a novel class of rate less codes that makes possible to control the computational complexity of Network Coding (NC). NC increases throughput of the networks via packet recombinations at the network nodes. In a NC scenario based on rate less codes, the recombinations at the nodes alter the packet degree distribution selected at the source and increase the computational complexity of the packet decoding process. Unlike other classes of rate less codes, BC preserve the degree distribution of the encoded packets through the recombinations at the nodes. Furthermore, BC enable to control the decoding complexity of each network node independently from the rest of the network. We evaluate BC in a P2P scenario using a purposely designed random-push protocol for live video streaming. The experiments show that BC achieve high encoding efficiency, enable nodes with different computational capabilities to coexist within the same network and reduce the processor load on a real mobile device by nearly 50%.

Rateless codes are designed to decode all the input symbols when a certain number of coded symbols have been received. However, it is possible to recover a subset of the input symbols from the actually received coded symbols: this process is called partial decoding and the number of recovered input symbols is termed the intermediate performance of rateless codes. In this paper we study the problem of the optimality of the partial decoding process: we say that a partial decoding algorithm is optimal if, given a rateless code, it is able to maximize the intermediate performance of the code, i.e. it is able to retreive the maximum number of input symbols when a certain number n of coded symbols have been received, for every n. We propose OPD, an optimal partial decoding algorithm for any rateless code, proving its optimality. The proposed algorithm is finally used to analyze the intermediate performance of LT codes.