It is well known that one of the most dangerous attacks against space transmissions is jamming. In particular, space telemetry downlinks are subject to extremely tight power and bandwidth limitations and are hence vulnerable to both accidental and intentional interference. While the Consultative Committee for Space Data Systems (CCSDS) has already standardized Low-Density Parity-Check (LDPC) codes for the use in space telemetry communications, we show that their performance is significantly deteriorated by the impact of pulsed jamming. We investigate the use of spatially coupled LDPC (SC-LDPC) codes as a promising alternative for improving reliability of telemetry channels in interference-heavy environments. We perform Monte Carlo simulations to assess the sensitivity of SC-LDPC-coded deep space communications to pulsed jamming, comparing attainable error rate performance with an LDPC code recommended in the CCSDS standard for telemetry synchronization and channel coding issues. Our results show that SC-LDPC codes can achieve improved performance, in terms of reduced bit error rate, under different pulsed jamming conditions.

Quantum low-density parity-check (QLDPC) codes provide a practical balance between error-correction capability and implementation complexity in quantum error correction (QEC). In this paper, we propose an algebraic construction based on dyadic matrices for designing both classical and quantum LDPC codes. The method first generates classical binary quasi-dyadic LDPC codes whose Tanner graphs have girth 6. It is then extended to the Calderbank-Shor-Steane (CSS) framework, where the two component parity-check matrices are built to satisfy the compatibility condition required by the recently introduced CAMEL-ensemble quaternary belief propagation decoder. This compatibility condition ensures that all unavoidable cycles of length 4 are assembled in a single variable node, allowing the mitigation of their detrimental effects by decimating that variable node.

Cyclic simplex codes are usually deemed as impractical because of their extremely low rate and large codeword length for practical values of the code dimension. To address these limitations, researchers have resorted to punctured simplex codes, focusing on primitive polynomials and employing statistical analyses to investigate their properties. This paper delves deeper into the properties of punctured binary simplex codes, also focusing on the recently introduced family of Primitive Rate-Compatible Low-Density Parity-Check (PRC-LDPC) codes. We study the average behavior of punctured simplex codes in terms of minimum distance properties. Furthermore, our results highlight the potential of high-rate PRC-LDPC codes to reach or even surpass the performance of state-of-the-art code families. We show how to design good codes by applying puncturing and shortening operations to cyclic simplex codes, also considering complexity-constrained scenarios.

Spatially coupled low-density parity-check (SC-LDPC) codes have recently attracted a lot of attention because of their optimal asymptotic performance. Time-invariant and periodically time-varying SC-LDPC codes are particularly interesting code families, since they have good finite-length performance and allow for a very compact representation. However, one of the key challenges to make SC-LDPC codes practical consists in finding efficient techniques for their encoding. In this paper, we propose a method to construct polynomial generator matrices for time-invariant and periodically time-varying SC-LDPC codes, based on their polynomial parity-check matrix. We show that it is always possible to find a polynomial generator matrix in quasi-standard form, where the systematic part consists of a diagonal matrix with identical entries, enabling efficient and non-catastrophic encoding. We also show that it is always possible to compute an equivalent rational function generator matrix in standard form that guarantees recursive, systematic (and therefore also non-catastrophic) encoding. Finally, to illustrate the method, we introduce a family of time-invariant SC-LDPC codes characterized by a binary parity-check matrix associated to a Tanner graph with girth larger than 4 and by good asymptotic and finite length performance. These codes naturally admit a quasi-standard polynomial generator matrix that, depending on the number of control symbols per period, either can be efficiently converted into polynomial standard form or remains in quasi-standard form but is relatively sparse. Both these features enable fast and efficient encoding.

The use of blockchains for data certification and traceability is now well established in both the literature and practical applications. However, while blockchain-based certification of individual data is clear and straightforward, the use of blockchain to certify large amounts of data produced on a nearly continuous basis still poses some challenges. In such a case, in fact, it is first necessary to collect the data in an off-chain buffer, and then to organize it, e.g., via Merkle trees, in order to keep the size and quantity of certification data to be written to the blockchain small. In this paper, we consider a typical system for blockchain-based traceability of a production process, and propose and comparatively analyze some strategies for certifying the data of such a process on blockchain, while maintaining the possibility of verifying their certification in a decentralized way.

Spatially Coupled Low-Density Parity-Check (SCLDPC) codes are characterized by very long codeword lengths. For this reason, they are usually decoded with sliding window algorithms, which allow piecewise processing and decoding of the codeword symbols. In order to mitigate error propagation, it is possible to adapt strategies, such as non-uniform window sizes and node doping. In this paper, we propose a novel adaptive decoding schedule, which can be integrated with the aforementioned strategies. Numerical results confirm that the proposed approach can successfully detect error propagation events with more accuracy than conventional log-likelihood ratio-based approaches. Simulation results show that the error rate performance of time-invariant SC-LDPC codes significantly improves when the proposed strategies are adopted.

In the context of space communications, as per the recommendation from the Consultative Committee for Space Data Systems regarding TeleCommand synchronization and coding, the Communications Link Transmission Unit is composed of a start sequence, coded data, and a tail sequence, which might be optional depending on the employed error correcting code. The task of detecting the tail sequence must be handled along with that of decoding the codewords containing the transmitted data, and this poses some challenges. In this paper, we propose a machine learning model for recognizing the tail sequence based on the analysis of metrics calculated during decoding, when the transmission is coded with Low-Density Parity-Check (LDPC) codes. The model is trained on data produced by an iterative decoder, commonly used in LDPC decoding, with noisy (random) codewords or the noisy tail sequence as inputs. We report the results of some preliminary experiments showing that this approach is capable of achieving very high levels of accuracy using multiple classifiers.

In this paper we study a special family of Low-Density Parity-Check (LDPC) codes, called Primitive Rate-Compatible (PRC) LDPC codes, obtained by applying puncturing to some properly chosen simplex codes. We investigate their minimum distance properties and give insights on the design of PRC-LDPC codes for use in scenarios where constraints on hardware and software resources require maintaining a low level of complexity. We numerically assess the error rate performance of these codes in such a setting, showing that they also demonstrate good performance under belief propagation decoding, in addition to low decoding complexity.

We introduce and study a family of rate-compatible Low-Density Parity-Check (LDPC) codes. The design of these codes starts from simplex codes, defined by parity-check matrices having a simple form stemming from the coefficients of a primitive polynomial. For this reason, we call the new codes Primitive Rate-Compatible LDPC (PRC-LDPC) codes. By applying puncturing to these codes, we obtain a bit-level granularity of the code rate. We show that, in order to achieve good LDPC codes, the underlying polynomials, besides being primitive, must meet some more stringent conditions with respect to those of classical punctured simplex codes. We leverage non-modular Golomb rulers to take these new requirements into account. We characterize the minimum distance properties of PRC-LDPC codes, and study and discuss their encoding and decoding complexity. Finally, we assess the error rate performance of high rate PRC-LDPC codes under iterative decoding.

Time-invariant spatially coupled low-density parity-check (TI-SC-LDPC) codes can be obtained by unwrapping quasi-cyclic (QC) LDPC block codes. This results in a free distance that is lower bounded by the minimum distance of the underlying QC-LDPC block codes. By introducing some variability in the syndrome former matrix, time-varying (TV) SC-LDPC codes are obtained, which trade an improved error correction performance for an increased decoding memory requirement and decoding complexity. A family of codes able to combine the advantages of TI-SC-LDPC codes with those of TV-SC-LDPC codes is that of periodically time-varying (PTV) SC-LDPC codes, based on a finite and periodic variation of the syndrome former matrix. In this paper we focus on such codes, and derive new upper bounds on the free distance of PTV-SC-LDPC code ensembles as well as on specific codes. By using these bounds, we show that PTV-SC-LDPC codes can achieve important improvements in the free distance over TI-SC-LDPC codes even using a very small period of variability, which corresponds to a minimal increase in memory and complexity. We also validate the new upper bounds through numerical experiments and assess the error correction performance of the corresponding codes through Monte Carlo simulations.

Cyber risk assessment is one of the top priorities of modern organizations and companies, owing to the massive amount of data they process on a daily basis and to the increasing number of successful cyber attacks. The probability of occurrence of these cyber incidents can be estimated by means of statistical tools, which exploit numerical categories to compute the probability that the organization will be breached by one or more cyber attacks. However, these approaches heavily rely on experts' estimates and/or on past data, which are not always available. In this paper we show that, by exploiting machine learning tools, cyber risk can be assessed by using some easily obtainable parameters (called maturity, complexity, attractiveness) representing the cyber posture of the organization under exam. To validate the method we propose, we apply it to three organizations in the healthcare sector having different values of maturity and complexity. The results highlight how the model can be successfully used to assign each organization a class of cyber risk, even in a crucial sector such as healthcare.

We propose a protocol to jointly achieve authentication and consensus on a blockchain network, in which endpoints are required to digitally sign some random message using fuzzy keys according to a classic fuzzy signature paradigm typical, for example, of biometric authentication. We consider classic RSA digital signatures, showing that fuzziness in the secret key translates into some noise affecting the derived signatures. The removal of such a noise provides the basis for building a blockchain consensus mechanism, which we name Proof of Fuzzy Signature (PoFS). It basically provides a special instance of Proof of Work in which the mining process corresponds to the de-noising process of RSA digital signatures derived from fuzzy keys. This way, the authentication process is delegated to a distributed network and, at the same time, requires executing the useful task of removing noise from fuzzy signatures.

In this paper we show that, when a binary primitive polynomial can be associated to a sparse Golomb ruler, the simplex code obtained by taking it as the code parity-check polynomial exhibits good distance properties and performance. We define some conditions under which the obtained codes are also Low-Density Parity-Check (LDPC) codes, and can hence be decoded through efficient iterative algorithms. We perform code puncturing, leading to a family of rate-adaptive codes, and we predict some of their structural properties in terms of minimum distance and weight distribution. We show that, in addition to having some useful properties, these codes achieve good performance in terms of error rate under LDPC decoding.

Time-varying spatially coupled low-density parity-check (SC-LDPC) codes with very large period are characterized by significantly better error rate performance and girth properties than their time-invariant counterparts, but the number of parameters they require to be described is usually very large and unpractical. Time-invariant SC-LDPC codes, which can be seen as periodically time-varying codes with unitary period, are represented through a small number of parameters and designed exploiting few degrees of freedom, but their error rate performance and girth properties are sub-optimal. In this paper, we show that the limits of time-invariant SC-LDPC codes can be overcome by transforming them into time-varying SC-LDPC codes with very small period. In particular, we show that periodically time-varying SC-LDPC codes with small period may exhibit significantly better girth properties than the corresponding time-invariant codes by exploiting a larger number of degrees of freedom in the code design, which however scale at most linearly with the product of the code period and the size of the considered base matrix.

We introduce a new family of linear block codes over $\mathbb{Z}_{q}$ that we name low-Lee-density parity-check (LLDPC) codes. These codes, which are embedded with the Lee metric, are characterized by a parity-check matrix whose rows and columns have low Lee weight. We propose general constructions of LLDPC codes and devise an efficient iterative decoding algorithm for them, with complexity that grows linearly with the code length. We assess the error rate performance of these codes through numerical simulations.

Iterative decoders used for decoding low-density parity-check (LDPC) and moderate-density parity-check (MDPC) codes are not characterized by a deterministic decoding radius and their error rate performance is usually assessed through intensive Monte Carlo simulations. However, several applications, like code-based cryptography, need guaranteed low values of the error rate, which are infeasible to assess through simulations, thus requiring the development of theoretical models for the error rate of these codes. Some models of this type already exist, but become computationally intractable for parameters of practical interest. Other approaches approximate the code ensemble behaviour through assumptions, which may not hold true for a specific code. We propose a theoretical analysis of the error correction capability of LDPC and MDPC codes that allows deriving tight bounds on the error rate at the output of parallel bit-flipping decoders. Special attention is devoted to the case of codes with small girth. Single-iteration decoding is investigated through a rigorous approach, which does not require any assumption and results in a guaranteed error correction capability for any single code. We show an example of application of the new bound to the context of code-based cryptography, where guaranteed error rates are needed to achieve strong security levels.

The error correction performance of low-density parity-check codes under iterative message-passing decoding is degraded by the presence of certain harmful objects existing in their Tanner graph representation. Depending on the context, such harmful objects are known as stopping sets, trapping sets, absorbing sets, or pseudocodewords. In this paper, we propose a general procedure, based on edge spreading, that enables the design of good quasi-cyclic spatially coupled low-density parity-check codes. These codes are derived from quasi-cyclic low-density parity-check (QC-LDPC) block codes and possess a significantly reduced multiplicity of harmful objects with respect to the original QC-LDPC block codes. The proposed procedure relies on a novel algorithm that greedily spans the search space of potential candidates to reduce the multiplicity of the target harmful objects. The effectiveness of the method is validated via examples and numerical computer simulations.

Differently from bounded-distance decoders used for algebraic codes, iterative decoders used for low-density parity-check (LDPC) codes are not characterized by a deterministic decoding radius. Therefore, the error rates of LDPC-coded transmissions are usually estimated heuristically through simulations. This is adequate for many applications like wireless communications, where a frame error rate (FER) in the order of 10-6or higher is usually targeted. However, lower values of FER can barely be assessed through simulations, and this limits the use of LDPC codes in applications requiring a lower FER, like optical communications and code-based cryptography. In this paper we introduce and study a version of the classic bit flipping (BF) decoder for which we are able to devise and develop a theoretical characterization of the FER. In addition, we consider a two-iteration hard-decision decoder for LDPC codes derived from BF, and discuss its error rate performance. Our results are validated through numerical simulations.

Time-varying spatially-coupled low-density parity-check convolutional codes (SC-LDPC-CCs) exhibit excellent features, but their representation requires a very large number of parameters. On the other hand, the description of time-invariant SC-LDPC-CCs is very convenient and their error rate performance, though usually worse, is often satisfactory. In this paper we investigate the girth properties of these codes, showing that the time-invariant ones have some weaknesses, which can be compensated by introducing a small periodicity in the code. By considering periodically time-varying codes, we achieve considerable improvements in the girth properties using few more degrees of freedom with respect to the time-invariant case.

Low decoding latency and complexity are two important requirements of channel codes used in many applications, like machine-to-machine communications. In this paper, we show how these requirements can be fulfilled by using some special quasi-cyclic low-density parity-check block codes and spatially coupled low-density parity-check convolutional codes that we denote as compact. They are defined by parity-check matrices designed according to a recent approach based on sequentially multiplied columns. This method allows obtaining codes with girth up to 12. Many numerical examples of practical codes are provided.

In this paper, time-invariant spatially coupled low-density parity-check convolutional codes (SC-LDPC-CCs) are considered, and the connections existing between their low-weight codewords and cycles in their Tanner graphs are studied. Using the polynomial representation of these codes, we show that parity-check matrices having columns with weight ≥2 can be analyzed considering a certain number of parity-check sub-matrices having regular columns with weight 2. These sub-matrices are associated to cycles in the code Tanner graph and define as many codes we denote as component codes. Based on this observation, we find that codewords of the main code can be expressed as combinations of codewords of the component codes. The design of codes free of codewords up to a certain weight is also addressed. We show that low-weight codewords in the main code can be avoided by removing some types of cycles in its Tanner graph. Our design approach is applied to some well-known ensembles of SC-LDPC-CCs to prove its effectiveness.

In this paper, we deal with time-invariant spatially coupled low-density parity-check convolutional codes (SC-LDPC-CCs). Classic design approaches usually start from quasi-cyclic low-density parity-check block codes and exploit suitable unwrapping procedures to obtain SC-LDPC-CCs. We show that the direct design of the SC-LDPC-CCs syndrome former matrix or, equivalently, the symbolic parity-check matrix, leads to codes with smaller syndrome former constraint lengths with respect to the best solutions available in the literature. We provide theoretical lower bounds on the syndrome former constraint length for the most relevant families of SC-LDPC-CCs, under constraints on the minimum length of cycles in their Tanner graphs. We also propose new code design techniques that approach or achieve such theoretical limits.

The writing operation of multi-level flash memories can suffer from voltage overshoots, which can be generally modeled as asymmetric errors of limited magnitude. Using suitable error correcting codes, these kinds of errors can be corrected. In particular, q-ary non-linear codes of length 2 are equivalent to packings of the plane modulo q with quasi-crosses. The design procedures for a number of such packings are presented.