This work tests the performance of Grover’s search circuits on some IBM superconducting quantum devices in case of the size of search space N = 16 and N = 32. Ideally, we expect to get an outcome probability distribution that is clearly peaked at the goal (marked) state. However, the quantum circuit executed on real devices is vulnerable to noise which leads to fluctuations in the results. The contributions of this paper are therefore the following: a) it presents two new Grover’s search circuits for N = 16 which were not yet reported in the state of the art; b) it shows performance difference between simulation results and results obtained on real devices; c) it shows the need of adding error correction on the circuit for N ≥ 32.

A new class of doubly-generalized LDPC codes over extended code alphabets has been recently proposed in [1]. These codes have been shown to have good asymptotic and minimum distance properties in the range of high code rates, and their structure is solely based on non-binary repetition and parity-check component codes. In this work we go further in the investigation of the new code class, called Repetition-parity-Parity (RPP) in the rest of the paper, by extending the definition of [1] to irregular degree distributions. One performs the asymptotic decoding analysis of the RPP ensemble over the binary erasure channel and formulates a degree distribution optimization problem by means of an approximated EXIT analysis. Some insights on the stability, minimum distance properties and low complexity encoding and decoding of RPP codes have been also developed.

This paper studies new geometrically shaped circular quadrature amplitude modulation (CQAM) constructions that can approach the Gaussian capacity. Then, we compare the performance of obtained constellations with existing Gaussian shaping strategies for regular and some simple optimized non-binary protographs. We also investigate the potential benefit of additional probabilistic shaping on the proposed CQAMs.

The maximum a posteriori (MAP) threshold corresponds to the fundamental limit that one can hope to achieve with the given channel code ensemble. Apart from theoretical interests, finding this limit is also desirable sincespatial-coupledcode ensembles approach this MAP threshold due to phenomenon termed asthreshold saturation. However finding this MAP threshold, in general, is known to be computationally prohibitive. This work proposes a tractable method for estimating the MAP threshold for various families of sparse-graph code ensembles over non-binary complex-input additive white Gaussian noise (AWGN) channel. Towards this, we provide a method to approximate the extended belief propagation generalized extrinsic information transfer (EBP-GEXIT) chart and estimate the MAP threshold by applying theMaxwell constructionto it. To illustrate the validity of our method, we study spatial coupling for serially-concatenated turbo-codes and numerically observe threshold saturation of these codes to the MAP thresholds estimated via our method.

In this paper, a new class of doubly-generalized LDPC codes is proposed. The particular point of the proposed construction is the presence of a small fraction of single parity-check codes at the variable nodes side. Together with the use of extended alphabets, the existence of such a fraction has been shown to improve the asymptotic decoding threshold, without harming the minimum distance behaviour of the code ensemble. Note that the improvement is more significant in cases where the code initially contains check nodes of high degrees, which corresponds to the region of moderate and high code rates.

We consider distributed and dynamic caching of coded content at small base stations (SBSs) in an area served by a macro base station (MBS). Specifically, content is encoded using a maximum distance separable code and cached according to a time-to-live (TTL) cache eviction policy, which allows coded packets to be removed from the caches at periodic times. Mobile users requesting a particular content download coded packets from SBSs within communication range. If additional packets are required to decode the file, these are downloaded from the MBS. We formulate an optimization problem that is efficiently solved numerically, providing TTL caching policies minimizing the overall network load. We demonstrate that distributed coded caching using TTL caching policies can offer significant reductions in terms of network load when request arrivals are bursty. We show how the distributed coded caching problem utilizing TTL caching policies can be analyzed as a specific single cache, convex optimization problem. Our problem encompasses static caching and the single cache as special cases. We prove that, interestingly, static caching is optimal under a Poisson request process, and that for a single cache the optimization problem has a surprisingly simple solution.

We investigate the problem of estimating the maximum a posteriori (MAP) threshold for serially concatenated turbo codes. First, we provide a method to compute this MAP threshold using a numerical approximation of the EBP-GEXIT chart and the Maxwell construction. Second, we explore where the spatially coupled belief propagation (BP) threshold is located with respect to the previously computed MAP threshold and analyze the saturation phenomenon of such schemes. Simulation results indicate that the BP threshold of the spatially coupled turbo-codes saturates to the MAP threshold obtained using the EBP-GEXIT chart.

We consider the caching of content in the mobile devices in a dense wireless network using maximum distance separable (MDS) codes. We focus on an area, served by a base station (BS), where mobile devices move around according to a random mobility model. Users requesting a particular file download the coded packets from caching devices within a communication range using device-to-device communication. If additional packets are required to decode the file, these are downloaded from the BS. We analyze the device mobility and derive a good approximation of the distribution of caching devices within the communication range of mobile devices at any given time. We then optimize the MDS codes to minimize the network load under a cache size constraint and show that using optimized MDS codes results in significantly lower network load compared to when caching the most popular files. We further show, numerically, that caching coded packets of each file on all mobile devices, i.e., maximal spreading, is optimal.

This work proposes a tractable evaluation of the maximum a posteriori (MAP) threshold of sparse-graph ensembles, by using an approximation for the extended belief propagation generalized extrinsic information transfer (EBP-GEXIT) function, first proposed by Measson et al. The approximation allows to find a MAP threshold in such numerically involved cases as the binary-input additive white Gaussian noise (AWGN) channel, graph ensembles with general component codes and/or irregularities. The paper contains examples of estimations of the MAP thresholds in the case of irregular low-density parity-check (LDPC), generalized LDPC, and doubly generalized LDPC codes ensembles. Our estimations are confirmed by numerical simulations.

We present the construction of a family of erasure correcting codes for distributed storage which achieve low repair bandwidth and complexity at the expense of a lower fault tolerance. The construction is based on two classes of codes, where the primary goal of the first class of codes is to provide fault tolerance, while the second class aims at reducing the repair bandwidth and repair complexity. The repair procedure is a two-step procedure where parts of the failed node are repaired in the first step using the first code. The downloaded symbols during the first step are cached in the memory and used to repair the remaining erased data symbols at minimal additional read cost during the second step. The first class of codes is based on maximum distance separable (MDS) codes modified using piggybacks, while the second class is designed to reduce the number of additional symbols that need to be downloaded to repair the remaining erased symbols. We numerically show that the proposed codes achieve better repair bandwidth compared to MDS codes, codes constructed using piggybacks, and local reconstruction/Pyramid codes, while a better repair complexity is achieved when compared to MDS, Zigzag, Pyramid codes, and codes constructed using piggybacks.

We consider the use of distributed storage (DS) to reduce the communication cost of content delivery in wireless networks. Content is stored (cached) in a number of mobile devices using an erasure correcting code. Users retrieve content from other devices using device-to-device communication or from the base station (BS), at the expense of higher communication cost. We address the repair problem when a device storing data leaves the cell. We introduce a repair scheduling where repair is performed periodically and derive analytical expressions for the overall communication cost of content download and data repair as a function of the repair interval. The derived expressions are then used to evaluate the communication cost entailed by DS using several erasure correcting codes. Our results show that DS can reduce the communication cost with respect to the case where content is downloaded only from the BS, provided that the repairs are performed frequently enough. If devices storing content arrive to the cell, the communication cost using DS is further reduced and, for a large enough arrival rate, it is always beneficial. Interestingly, we show that maximum distance separable codes, which do not perform well for classical DS, can yield a low overall communication cost in wireless DS.

We analyze the asymptotic performance of nonbinary spatially coupled low-density parity-check (SC-LDPC) code ensembles defined over the general linear group on the binary erasure channel. In particular, we prove the threshold saturation of belief propagation decoding to the so-called potential threshold, using the proof technique based on potential functions introduced by Yedla et al., assuming that the potential function exists. We rewrite the density evolution of nonbinary SC-LDPC codes in an equivalent vector recursion form which is suited for the use of the potential function. We then discuss the existence of the potential function for the general case of vector recursions defined by multivariate polynomials, and give a method to construct it. We define a potential function in a slightly more general form than the one by Yedla et al., in order to make the technique based on potential functions applicable to the case of nonbinary LDPC codes. We show that the potential function exists if a solution to a carefully designed system of linear equations exists. Furthermore, we numerically show the existence of a solution to the system of linear equations for a large number of nonbinary LDPC code ensembles, which allows us to define their potential function and thus prove threshold saturation.

In this paper we consider distributed allocation problems with memory constraint limits. Firstly, we propose a tractable relaxation to the problem of optimal symmetric allocations. The approximated problem is based on the Q-error function, and its solution approaches the solution of the initial problem, as the number of storage nodes in the network grows. Secondly, exploiting this relaxation, we are able to formulate and to solve the problem for storage allocations for memory-limited DSS storing and arbitrary memory profiles. Finally, we discuss the extension to the case of multiple data objects, stored in the DSS.

We present the construction of a new family of erasure correcting codes for distributed storage that yield low repair bandwidth and low repair complexity. The construction is based on two classes of parity symbols. The primary goal of the first class of symbols is to provide good fault tolerance, while the second class facilitates node repair, reducing the repair bandwidth and the repair complexity. We compare the proposed codes with other codes proposed in the literature.

Spatially coupled low-density parity-check (SC-LDPC) codes are considered for transmission over the block-fading channel. The diversity order of the SC-LDPC codes is studied using density evolution and simulation results. We demonstrate that the diversity order of the code can be increased, without lowering the code rate, by simply increasing the coupling parameter (memory) of a SC-LDPC code. For a (3,6)-regular SC-LDPC code with rate R = 1=2 and memory mcc= 4 a remarkable diversity of d = 10 is achieved without the need for any specific code structure. The memory of the SC-LDPC codes makes them robust against a non-stationary mobile-radio environment. The decoding of SC-LDPC codes using a latency constrained sliding window decoder is also considered.

The paper considers the impact of changing code parameters on the network load, for some given storage-flexible Data Center Network (DCN), i.e. such DCN in which the reliability and the storage volume can be modified during the storage life of the DCN data. Two regimes of the network load are considered: transition (during the migration process) and stationary (at the end of the migration process). Our main result is the derivation of the link between reliability and network load via code parameters; clearly, this link is code-dependent. Two different erasure-coding families are considered as examples (MDS and LDPC codes), to illustrate the dependence in two different coding cases.

The paper provides simple lower and upper bounds on block PBand bit Pbperformances of spatially-coupled LDPC (SC-LDPC) codes over a particular model of the block erasure channel. As expected, the spatial coupling structure helps in the correction of bursty erasures, and the decoding performance of SC-LDPC codes improves if the coupling parameter w increases.

We propose here a new construction of quantum codes combining an improved version of a family of spatially coupled quantum LDPC codes, suggested in [1], with a family of error reducing turbo-codes of [2]. This new construction displays outstanding performances under iterative decoding for noise levels very close to the hashing bound, without needing qubits, protected from noise as in [1].

We generalize a construction of non-binary quantum LDPC codes over F2mdue to [KHIK11] and apply it in particular to toric codes. We obtain in this way not only codes with better rates than toric codes but also improve dramatically the performance of standard iterative decoding. Moreover, the new codes obtained in this fashion inherit the distance properties of the underlying toric codes and have therefore a minimum distance which grows as the square root of the length of the code for fixed m.

In this paper, the asymptotic iterative performance of block-Markov, sparse-graph codes over the binary erasure relay channel is investigated. Note that the full-duplex relay channel is a particular case of the considered model. We obtain two interesting results: a) the block-Markov structure does not improve the asymptotic performance of good erasure-correcting sparse-graph codes; b) under certain conditions, it does however improve the asymptotic performance of good erasure-detecting (i.e. bad erasure-correcting) sparse-graph codes.

We propose here a new construction of spatially coupled quantum LDPC codes using a small amount of entangled qubit pairs shared between the encoder and the decoder which improves quite significantly all other constructions of quantum LDPC codes or turbo-codes with the same rate.

This paper deals with the design of low-rate sparse-graph codes, having a linear minimum distance d_{min} in the blocklength n. Its main contributions are: a) a necessary condition on a general family of sparse-graph codes with linear d_{min}; b) a justification of having degree-1 bits in the low-rate code structure; c) a new, efficient ensemble of low-rate sparse-graph codes with bits of degree 1, designed so that the necessary condition (a) is satisfied.

We study incremental redundancy hybrid automatic repeat request (IR-HARQ) schemes based on punctured, finite-length, low-density, parity-check (LDPC) codes. The transmission is assumed to take place over time-varying binary erasure channels, such as mobile wireless channels at the application layer. We analyze and optimize the throughput and delay performance of these IR-HARQ protocols under iterative, message-passing decoding. We derive bounds on the performance that are achievable by such schemes, and show that, with a simple extension, the iteratively decoded, punctured LDPC code-based IR-HARQ protocol can be made rateless and operating close to the general theoretical optimum for a wide range of channel erasure rates.

We study iterative receivers for joint decoding and channel-state estimation for transmission on block-fading channels of root- LDPC-coded signals. Root-LDPC codes are known to be most performant codes for block-fading channels, as their spacial "root" structure allows to get the full-diversity property. This property ensures a good error decoding performance of root LDPC codes, especially in contrast with the performance standard LDPC codes (having the maximum diversity equal to 1). However, as any channel code, root-LDPC codes also suffer from the diversity loss when the channel state information is not known at the receiver. In this work we propose a joint channel estimation- decoding scheme for root- LDPC codes that helps to overcome this problem and still to have the full-diversity.

The asymptotic iterative performance of the block-Markov encoding scheme, defined over bilayer LDPC codes, is analyzed. This analysis is carried out for both half-duplex and full-duplex regimes. For the sake of clarity and simplicity, a transmission over the binary erasure relay channel is assumed. To analyze the iterative performance of the coding scheme, the asymptotic threshold boundary γ (∈1, ∈2) is used as a performance measure. It is derived for two sparse-graph ensembles: Block-Markov Bilayer-Expurgated and -Lengthened LDPC ensembles.

In this paper, the transmission over the binary erasure channel (BEC) using non-binary LDPC (NBLDPC) codes is considered. The concept of peeling decoder and stopping sets is generalized to NBLDPC codes. Using these generalizations, a combinatorial characterization of decoding failures of NBLDPC codes is given, under assumption that the Belief Propagation (BP) decoder is used. Then, the residual ensemble of codes resulted by the BP decoder is defined and the design rate and the expectation of total number of codewords of the residual ensemble are computed. The decoding failure criterion combined with the density evolution analysis helps us to compute the asymptotic residual degree distribution for NBLDPC codes. Our approach to compute the residual degree distribution on the check node side is not efficient as it is based on enumeration of all the possible connections on the check node side which satisfy the decoding failure criterion. So, the computation of the asymptotic check node side residual degree distribution and further part of our analysis is performed for NBLDPC codes over GF2mwithm= 2 . In order to show that asymptotically almost every code in such LDPC ensemble has a rate equal to the design rate, we generalize the argument of the Maxwell construction to NBLDPC codes, defined over GF22. It is also observed that, like in the binary setting, the Maxwell construction, relating the performance of MAP and BP decoding holds in this setting.

This paper provides a performance analysis of the regular (c, d) LDPC code ensemble of codelength n with parity-check matrices defined over the general linear group GL(2m). The transmission is assumed to take place over the binary erasure channel with erasure probability ε. In this work, the scaling approximation of the block erasure rate is generalized to the non-binary case, and the scaling parameter α of the approximation is derived. The proposed estimation is then compared with numerical results, showing that it predicts well the slope of the block erasure rate vs. channel erasure probability.

This paper is the first part of an investigation if the capacity of a binary-input memoryless symmetric channel under ML decoding can be achieved asymptotically by using non-binary LDPC codes. We consider (l, r)-regular LDPC codes both over finite fields and over the general linear group and compute their asymptotic binary weight distributions in the limit of large blocklength and of large alphabet size. A surprising fact, the average binary weight distributions that we obtain do not tend to the binomial one for values of normalized binary weights ¿ smaller than 1-2-l/r. However, it does not mean that non-binary codes do not achieve the capacity asymptotically, but rather that there exists some exponentially small fraction of codes in the ensemble, which contains an exponentially large number of codewords of poor weight. The justification of this fact is beyond the scope of this paper and will be given in.

This paper is devoted to the analysis of finite-length iterative performance of punctured LDPC ensembles in the waterfall region, assuming the transmission over the binary erasure channel (BEC). The analysis is carried out using the scaling approach proposed in. Two punctured ensembles are considered: (a) randomly punctured ensembles, in the sense that each bit of a codeword is punctured with some puncturing probability; (b) ensembles with a fixed puncturing fraction of bits of each degree. In both cases, parameters of the scaling approximation are completely determined in terms of the ensemble parameters such as left, right and puncturing degree distributions.

A possibility of estimating the finite-length performance of sparse-graph code ensembles gives two opportunities: to compare different codes of the same length in a context very close to real, practical applications and to perform the parameter optimization for a given code length [2]. We need a finite-length approximation that is valid for any code ensemble. The scaling approach seems to be a tool, general enough to provide such an approximation. However, the analytical derivation of parameters of the scaling approximation has been successful only for LDPC codes [1]; despite several attempts [25], [20], no such result was proposed for other code ensembles. In this paper, we focus on the finite-length performance of turbo-like codes, by applying the scaling approach to this case. In particular, by assuming the transmission over the binary erasure channel, we conjecture the scaling law and derive its scaling parameter. As examples, we present the performance estimation for Repeat-Accumulate codes [11], parallel turbo codes [8] and TLDPC codes [5], in all cases matching well the numerical results.

In this paper we consider the finite-length performance of turbo codes in the waterfall region assuming the transmission on the binary erasure channel. We extend the finite-length scaling law of LDPC codes and of repeat-accumulate codes to this ensemble and we derive the scaling parameter. The obtained performance estimations match very well with numerical results.

We generalize the results about how to compute iterative decoding thresholds over the binary erasure channel of non-binary LDPC code ensembles of [16] to non-binary TLDPC codes [2], [3]. We show in this case how density evolution can be performed in order to calculate iterative decoding thresholds and find several families with a very simple regular structure and thresholds close to the Shannon limit. To check the performances of these codes over other channels we have tested one of the simplest codes over F4which has rate 1/2 on the Gaussian channel. For the (binary) length 1008 for instance, without any optimization on the permutation structure of the code, it matches the performances of the best binary codes of the same length up to the word-error rate 10-3. We also notice that all LDPC codes (binary or not) having at least two symbols of degree 2 per parity-check equation can be represented as a special kind of TLDPC codes. We show that this representation and the associated decoding algorithm leads in the case of cycle codes to a significant reduction of the number of iterations which are needed for iterative decoding.

We study necessary conditions which have to be satisfied in order to have LDPC codes with linear minimum distance. We give two conditions of this kind in this paper. These conditions are not met for several interesting code families: this shows that they are not asymptotically good. The second one concerns LDPC codes that have a Tanner graph in which there are cycles linking variable nodes of degree 2 together and provides some insight about the combinatorial structure of some low-weight codewords in such a case. When the LDPC code family is obtained from the lifts of a given protograph and if there are such cycles in the protograph, the second condition seems to capture really well the linear minimum distance character of the code. This is illustrated by a code family which is asymptotically good for which there is a cycle linking all the variable nodes of degree 2 together. Surprisingly, this family is only a slight modification of a family which does not satisfy the second condition.

We investigate a new class of codes which is in a sense a hybrid between LDPC codes and turbo-codes. Some members of this new class have been shown to be asymptotically good and we conjecture that such a behavior holds for all classes of codes presented here. They all display excellent iterative decoding performances with no error floor at block error rates up to 10-6 for lengths of several thousand together with low average decoding complexity.

We investigate a new class of codes which is in some senses a hybrid between LDPC codes and turbo-codes. We show that when the parameters of this class are well chosen they have very good iterative decoding performances and at the same time a minimum distance which is typically linear in the code length