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Let $a$ and $b$ be two non-zero elements of a finite field $\mathbb{F}_q$, where $q>2$. It has been shown that if $a$ and $b$ have the same multiplicative order in $\mathbb{F}_q$, then the families of $a$-constacyclic and $b$-constacyclic codes over $\mathbb{F}_q$ are monomially equivalent. In this paper, we investigate the monomial equivalence of $a$-constacyclic and $b$-constacyclic codes when $a$ and $b$ have distinct multiplicative orders. We present novel conditions for establishing monomial equivalence in such constacyclic codes, surpassing previous methods of determining monomially equivalent constacyclic and cyclic codes. As an application, we use these results to search for new linear codes more systematically. In particular, we present more than $70$ new record-breaking linear codes over various finite fields, as well as new binary quantum codes., Comment: 15 pages, 4 figures, 2 tables

The machinery of industrial environments was connected to the Internet years ago with the scope of increasing their performance. However, this change made such environments vulnerable against cyber-attacks that can compromise their correct functioning resulting in economic or social problems. Moreover, implementing cryptosystems in the communications between operational technology (OT) devices is a more challenging task than for information technology (IT) environments since the OT networks are generally composed of legacy elements, characterized by low-computational capabilities. Consequently, implementing cryptosystems in industrial communication networks faces a trade-off between the security of the communications and the amortization of the industrial infrastructure. Critical Infrastructure (CI) refers to the industries which provide key resources for the daily social and economical development, e.g. electricity. Furthermore, a new threat to cybersecurity has arisen with the theoretical proposal of quantum computers, due to their potential ability of breaking state-of-the-art cryptography protocols, such as RSA or ECC. Many global agents have become aware that transitioning their secure communications to a quantum secure paradigm is a priority that should be established before the arrival of fault-tolerance. In this paper, we aim to describe the problematic of implementing post-quantum cryptography (PQC) to CI environments. For doing so, we describe the requirements for these scenarios and how they differ against IT. We also introduce classical cryptography and how quantum computers pose a threat to such security protocols. Furthermore, we introduce state-of-the-art proposals of PQC protocols and present their characteristics. We conclude by discussing the problematic of integrating PQC in industrial environments., Comment: 27 pages, 7 figures, 10 tables

Quantum technologies have the potential to solve certain computationally hard problems with polynomial or super-polynomial speedups when compared to classical methods. Unfortunately, the unstable nature of quantum information makes it prone to errors. For this reason, quantum error correction is an invaluable tool to make quantum information reliable and enable the ultimate goal of fault-tolerant quantum computing. Surface codes currently stand as the most promising candidates to build near term error corrected qubits given their two-dimensional architecture, the requirement of only local operations, and high tolerance to quantum noise. Decoding algorithms are an integral component of any error correction scheme, as they are tasked with producing accurate estimates of the errors that affect quantum information, so that they can subsequently be corrected. A critical aspect of decoding algorithms is their speed, since the quantum state will suffer additional errors with the passage of time. This poses a connundrum, where decoding performance is improved at the expense of complexity and viceversa. In this review, a thorough discussion of state-of-the-art decoding algorithms for surface codes is provided. The target audience of this work are both readers with an introductory understanding of the field as well as those seeking to further their knowledge of the decoding paradigm of surface codes. We describe the core principles of these decoding methods as well as existing variants that show promise for improved results. In addition, both the decoding performance, in terms of error correction capability, and decoding complexity, are compared. A review of the existing software tools regarding surface codes decoding is also provided, Comment: 59 pages, 38 figures

Quantum channel capacity is a fundamental quantity in order to understand how good can quantum information be transmitted or corrected when subjected to noise. However, it is generally not known how to compute such quantities, since the quantum channel coherent information is not additive for all channels, implying that it must be maximized over an unbounded number of channel uses. This leads to the phenomenon known as superadditivity, which refers to the fact that the regularized coherent information of $n$ channel uses exceeds one-shot coherent information. In this article, we study how the gain in quantum capacity of qudit depolarizing channels relates to the dimension of the systems considered. We make use of an argument based on the no-cloning bound in order to proof that the possible superadditive effects decrease as a function of the dimension for such family of channels. In addition, we prove that the capacity of the qudit depolarizing channel coincides with the coherent information when $d\rightarrow\infty$. We also discuss the private classical capacity and obain similar results. We conclude that when high dimensional qudits experiencing depolarizing noise are considered, the coherent information of the channel is not only an achievable rate but essentially the maximum possible rate for any quantum block code., Comment: 10 pages, 2 figures

The minimum weight perfect matching (MWPM) decoder is the standard decoding strategy for quantum surface codes. However, it suffers a harsh decrease in performance when subjected to biased or non-identical quantum noise. In this work, we modify the conventional MWPM decoder so that it considers the biases, the non-uniformities and the relationship between $X$, $Y$ and $Z$ errors of the constituent qubits of a given surface code. Our modified approach, which we refer to as the recursive MWPM decoder, obtains an $18\%$ improvement in the probability threshold $p_{th}$ under depolarizing noise. We also obtain significant performance improvements when considering biased noise and independent non-identically distributed (i.ni.d.) error models derived from measurements performed on state-of-the-art quantum processors. In fact, when subjected to i.ni.d. noise, the recursive MWPM decoder yields a performance improvement of $105.5\%$ over the conventional MWPM strategy and, in some cases, it even surpasses the performance obtained over the well-known depolarizing channel.

Recent experimental studies have shown that the relaxation time ($T_1$) and the dephasing time ($T_2$) of superconducting qubits fluctuate considerably over time. To appropriately consider this time-varying nature of the $T_1$ and $T_2$ parameters, a new class of quantum channels, known as Time-Varying Quantum Channels (TVQCs), has been proposed. In previous works, realizations of multi-qubit TVQCs have been assumed to be equal for all the qubits of an error correction block, implying that the random variables that describe the fluctuations of $T_1$ and $T_2$ are block-to-block uncorrelated, but qubit-wise perfectly correlated for the same block. Physically, the fluctuations of these decoherence parameters are explained by the incoherent coupling of the qubits with unstable near-resonant two-level-systems (TLS), which indicates that such variations may be local to each of the qubits of the system. In this article, we perform a correlation analysis of the fluctuations of the relaxation times of multi-qubit quantum processors ibmq\_quito, ibmq\_belem, ibmq\_lima, ibmq\_santiago and ibmq\_bogota. Our results show that it is reasonable to assume that the fluctuations of the relaxation and dephasing times of superconducting qubits are local to each of the qubits of the system. Based on these results, we discuss the multi-qubit TVQCs when the fluctuations of the decoherence parameters for an error correction block are qubit-wise uncorrelated (as well as from block-to-block), a scenario we have named the Fast Time-Varying Quantum Channel (FTVQC). Furthermore, we lower bound the quantum capacity of general FTVQCs based on a quantity we refer to as the ergodic quantum capacity. Finally, we use numerical simulations to study the performance of quantum error correction codes (QECC) when they operate over FTVQCs., Comment: 21 pages

Surface codes are generally studied based on the assumption that each of the qubits that make up the surface code lattice suffers noise that is independent and identically distributed (i.i.d.). However, real benchmarks of the individual relaxation ($T_1$) and dephasing ($T_2$) times of the constituent qubits of state-of-the-art quantum processors have recently shown that the decoherence effects suffered by each particular qubit actually vary in intensity. In consequence, in this article we introduce the independent non-identically distributed (i.ni.d.) noise model, a decoherence model that accounts for the non-uniform behaviour of the docoherence parameters of qubits. Additionally, we use the i.ni.d model to study how it affects the performance of a specific family of Quantum Error Correction (QEC) codes known as planar codes. For this purpose we employ data from four state-of-the-art superconducting processors: ibmq\_brooklyn, ibm\_washington, Zuchongzhi and Rigetti Aspen-M-1. Our results show that the i.i.d. noise assumption overestimates the performance of surface codes, which can suffer up to $95\%$ performance decrements in terms of the code pseudo-threshold when they are subjected to the i.ni.d. noise model. Furthermore, we consider and describe two methods which enhance the performance of planar codes under i.ni.d. noise. The first method involves a so-called re-weighting process of the conventional minimum weight perfect matching (MWPM) decoder, while the second one exploits the relationship that exists between code performance and qubit arrangement in the surface code lattice. The optimum qubit configuration derived through the combination of the previous two methods can yield planar code pseudo-threshold values that are up to $650\%$ higher than for the traditional MWPM decoder under i.ni.d. noise., Comment: Includes supplementary material (19 pages total)

Recent experimental studies have shown that the relaxation time, $T_1$, and the dephasing time, $T_2$, of superconducting qubits fluctuate considerably over time. Time-varying quantum channel (TVQC) models have been proposed in order to consider the time varying nature of the parameters that define qubit decoherence. This dynamic nature of quantum channels causes a degradation of the performance of quantum error correction codes (QECC) that is portrayed as a flattening of their error rate curves. In this article, we introduce the concepts of quantum outage probability and quantum hashing outage probability as asymptotically achievable error rates by a QECC with quantum rate $R_Q$ operating over a TVQC. We derive closed-form expressions for the family of time-varying amplitude damping channels (TVAD) and study their behaviour for different scenarios. We quantify the impact of time-variation as a function of the relative variation of $T_1$ around its mean. We conclude that the performance of QECCs is limited in many cases by the inherent fluctuations of their decoherence parameters and corroborate that parameter stability is crucial to maintain the excellent performance observed over static quantum channels., Comment: 11 pages, 4 figures, will be submitted to Quantum

The quantum paradigm presents a phenomenon known as degeneracy that should improve the performance of quantum error correcting codes. However, the effects of this mechanism are sometimes ignored when evaluating the performance of sparse quantum codes and the logical error rate is not always correctly reported. In this paper, we discuss previously existing methods to compute the logical error rate and we present an efficient coset-based method inspired by classical coding strategies to estimate degenerate errors. Additionally, we show that the proposed method presents a computational advantage for the family of Calderbank-Shor-Steane codes. We use this method to prove that degenerate errors are frequent in a specific family of sparse quantum codes, which stresses the importance of accurately reporting their performance. Our results also reveal that the modified decoding strategies proposed in the literature are an important tool to improve the performance of sparse quantum codes., Comment: 11 pages, 1 figure. Updated to include discussion on computational complexity

We describe the design and show first results of a large solid angle X-ray emission spectrometer that is optimized for energies between 1.5 keV and 5.5 keV. The spectrometer is based on an array of 11 cylindrically bent Johansson crystal analyzers arranged in a non-dispersive Rowland circle geometry. The smallest achievable energy bandwidth is smaller than the core hole lifetime broadening of the absorption edges in this energy range. Energy scanning is achieved using an innovative design, maintaining the Rowland circle conditions for all crystals with only four motor motions. The entire spectrometer is encased in a high-vacuum chamber that allocates a liquid helium cryostat and provides sufficient space for in situ cells and operando catalysis reactors., Comment: Manuscript (11 pages, 10 figures) plus supporting information (8 pages, 12 figures)

Industry 4.0 applications foster new business opportunities but they also pose new and challenging requirements, such as low latency communications and highly reliable systems. They enable to exploit novel wireless technologies (5G), but it would also be crucial to rely on architectures that appropriately support them. Thus, the combination of fog and cloud computing is emerging as one potential solution. It can dynamically allocate the workload depending on the specific needs of each application. Our main goal is to provide a highly reliable and dynamic architecture, which minimizes the time that an end node or user, for instance a car in a smart mobility application, spends in downloading the required data. In order to achieve this, we have developed an optimal distribution algorithm that decides, based on multiple parameters of the proposed system model, the amount of information that should be stored at, or retrieved from, each node to minimize the data download time. Our scheme exploits Network Coding (NC) as a tool for data distribution, as a key enabler of the proposed solution. We compare the performance of our proposed scheme with other alternative solutions, and the results show that there is a clear gain in terms of the download time., Comment: Funding: the Basque Government through the ADDISEND (KK-2018/00115) and DIGITAL (KK-2019/00095) Elkartek programs, the H2020 research framework under the ELASTIC project (825473), and the Spanish Ministry of Economy and Competitiveness through CARMEN (TEC2016-75067-C4-3-R), ADVICE (TEC2015-71329-C2-1-R), FIERCE (RTI2018-093475-A-100) and COMONSENS (TEC2015-69648-REDC) projects

This letter introduces a novel channel coding design framework for short-length codewords that permits balancing the tradeoff between the bit error rate floor and waterfall region by modifying a single real-valued parameter. The proposed approach is based on combining convolutional coding with a $q$-ary linear combination and unequal energy allocation, the latter being controlled by the aforementioned parameter. EXIT charts are used to shed light on the convergence characteristics of the associated iterative decoder, which is described in terms of factor graphs. Simulation results show that the proposed scheme is able to adjust its end-to-end error rate performance efficiently and easily, on the contrary to previous approaches that require a full code redesign when the error rate requirements of the application change. Simulations also show that, at mid-range bit-error rates, there is a small performance penalty with respect to the previous approaches. However, the EXIT chart analysis and the simulation results suggest that for very low bit-error rates the proposed system will exhibit lower error floors than previous approaches., Comment: 4 pages; submitted to IEEE Communications Letters

This paper proposes a novel joint non-binary network-channel code for the Time-Division Decode-and-Forward Multiple Access Relay Channel (TD-DF-MARC), where the relay linearly combines -- over a non-binary finite field -- the coded sequences from the source nodes. A method based on an EXIT chart analysis is derived for selecting the best coefficients of the linear combination. Moreover, it is shown that for different setups of the system, different coefficients should be chosen in order to improve the performance. This conclusion contrasts with previous works where a random selection was considered. Monte Carlo simulations show that the proposed scheme outperforms, in terms of its gap to the outage probabilities, the previously published joint network-channel coding approaches. Besides, this gain is achieved by using very short-length codewords, which makes the scheme particularly attractive for low-latency applications., Comment: 28 pages, 9 figures; Submitted to IEEE Journal on Selected Areas in Communications - Special Issue on Theories and Methods for Advanced Wireless Relays, 2012