Rank Metric Research Articles

Algebraic Geometry Codes in the Sum–Rank Metric

Algebraic Geometry Codes in the Sum–Rank Metric

Saturating linear sets of minimal rank

Saturating sets are combinatorial objects in projective spaces over finite fields that have been intensively investigated in the last three decades. They are related to the so-called covering problem of codes in the Hamming metric. In this paper, we consider the recently introduced linear version of such sets, which is, in turn, related to the covering problem in the rank metric. The main questions in this context are how small the rank of a saturating linear set can be and how to construct saturating linear sets of small rank. Recently, Bonini, Borello, and Byrne provided a lower bound on the rank of saturating linear sets in a given projective space, which is shown to be tight in some cases. In this paper, we provide construction of saturating linear sets meeting the lower bound and we develop a link between the saturating property and the scatteredness of linear sets. The last part of the paper is devoted to show some parameters for which the bound is not tight.

On the generalised rank weights of quasi-cyclic codes

<p style='text-indent:20px;'>Generalised rank weights were formulated in analogy to Wei's generalised Hamming weights, but for the rank metric. In this paper we study the generalised rank weights of quasi-cyclic codes, a special class of linear codes usually studied for their properties in error correction over the Hamming metric. By using the algebraic structure of quasi-cyclic codes, a new upper bound on the generalised rank weights of quasi-cyclic codes is formulated, which is tighter than the known Singleton bound. Additionally, it is shown that the first generalised rank weight of self-dual <inline-formula><tex-math id="M1">\begin{document}$ 1 $\end{document}</tex-math></inline-formula>-generator quasi-cyclic codes is almost completely determined by the choice of <inline-formula><tex-math id="M2">\begin{document}$ {\mathbb F}_{q^{m}} $\end{document}</tex-math></inline-formula>.</p>

Empirical Analysis of Decision Recommendation Models for Various Processes from A Pragmatic Perspective

Decision recommendation models allow researchers and process designers to identify & implement high-efficiency processes under ambiguous situations. These models perform multiparametric analysis on the given process sets in order to recommend high quality decisions that assist in improving process-based efficiency levels. A wide variety of models are proposed by researchers for implementation of such recommenders, and each of them varies in terms of their functional nuances, applicative advantages, internal operating characteristics, contextual limitations, and deployment-specific future scopes. Thus, it is difficult for researchers and process designers to identify optimal models for their functionality-specific use cases. Due to which, they tend to validate multiple process models, which increases deployment time, cost & complexity levels.To overcome this ambiguity, a detailed survey of different decision process recommendation models is discussed in this text. It was observed that Fuzzy Logic, Analytical Hierarchical Processing (AHP), Technique for Order Performance by Similarity to Ideal Solution (TOPSIS), and their variants are highly useful for recommendation of efficient decisions. Based on this survey, readers will be able to identify recently proposed decision recommendation models, and identify functionality-specific models for their deployments. To further assist the model selection process, this text compares the reviewed models in terms of their computational complexity, efficiency of recommendation, delay needed for recommendation, scalability and contextual accuracy levels. Based on this comparison, readers will be able to identify performance-specific models for their deployments. This text also proposes evaluation of a novel Decision Recommendation Rank Metric (DRRM), which combines these parameters, in order to identify models that can optimally perform w.r.t. multiple process metrics. Referring to this parameter comparison, readers will be able to identify optimal recommendation models for enhancing performance of their decision recommendations under real-time scenarios.

Statistical Review Evaluation of 5G Antenna Design Models from a Pragmatic Perspective under Multi-Domain Application Scenarios

Antenna design for the 5G spectrum requires analysis of contextual frequency bands, design of miniaturization techniques, gain improvement models, polarization techniques, standard radiation pattern designs, metamaterial integration, and substrate selection. Most of these models also vary in terms of qualitative & and quantitative parameters, which include forward gain levels, reverse gain, frequency response, substrate types, antenna shape, feeding levels, etc. Due to such a wide variety in performance, it is ambiguous for researchers to identify the optimum models for their application-specific use cases. This ambiguity results in validating these models on multiple simulation tools, which increases design delays and the cost of deployments. To reduce this ambiguity, a survey of recently proposed antenna design models is discussed in this text. This discussion recommended that polarization optimization and gain maximization are the major impact factors that must be considered while designing antennas. It is also recommended that collocated microstrip slot antennas, fully planar dual-polarized broadband antennas, and real-time deployments of combined slot antenna pairs with wide-band decoupling are very advantageous. Based on this discussion, researchers will be able to identify optimal performance-specific models for different applications. This discussion also compares underlying models in terms of their quantitative parameters, which include forward gain levels, bandwidth, complexity of deployment, scalability, and cost metrics. Upon referring to this comparison, researchers will be able to identify the optimum models for their performance-specific use cases. This review also formulates a novel Antenna Design Rank Metric (ADRM) that combines the evaluated parameters, thereby allowing readers to identify antenna design models that are optimized for multiple parameters and can be used for large-scale 5G communication scenarios.

Saturating systems and the rank-metric covering radius

We introduce the concept of a rank-saturating system and outline its correspondence to a rank-metric code with a given covering radius. We consider the problem of finding the value of s_{q^m/q}(k,rho ), which is the minimum mathbb {F}_q-dimension of a q-system in mathbb {F}_{q^m}^k that is rank-rho -saturating. This is equivalent to the covering problem in the rank metric. We obtain upper and lower bounds on s_{q^m/q}(k,rho ) and evaluate it for certain values of k and rho . We give constructions of rank-rho -saturating systems suggested from geometry.

Group von Neumann Algebras, Inner Amenability, and Unit Groups of Continuous Rings

Abstract We prove that, if a discrete group $G$ is not inner amenable, then the unit group of the ring of operators affiliated with the group von Neumann algebra of $G$ is non-amenable with respect to the topology generated by its rank metric. This provides examples of non-discrete irreducible, continuous rings (in von Neumann’s sense) whose unit groups are non-amenable with regard to the rank topology. Our argument establishes and uses connections with Eymard–Greenleaf amenability of the action of the unitary group of a $\textrm{II}_{1}$ factor on the associated space of projections of a fixed trace.

Rank Metric Codes and Their Galois Duality

Rank Metric Codes and Their Galois Duality

The Partial-Inverse Approach to Linearized Polynomials and Gabidulin Codes With Applications to Network Coding

This paper introduces the partial-inverse problem for linearized polynomials and develops its application to decoding Gabidulin codes and lifted Gabidulin codes in linear random network coding. The proposed approach is a natural generalization of its counterpart for ordinary polynomials, thus providing a unified perspective on Reed–Solomon codes for the Hamming metric and for the rank metric. The basic algorithm for solving the partial-inverse problem is a common parent algorithm of a Berlekamp–Massey algorithm, a Euclidean algorithm, and yet another algorithm, all of which are obtained as easy variations of the basic algorithm. Decoding Gabidulin codes can be reduced to the partial-inverse problem via a key equation with a new converse. This paper also develops new algorithms for interpolating crisscross erasures and for joint decoding of errors, erasures, and deviations in random network coding.

Degenerate flag varieties in network coding

<p style='text-indent:20px;'>Building upon the application of flags to network coding introduced in [<xref ref-type="bibr" rid="b6">6</xref>], we develop a variant of this coding technique that uses degenerate flags. The information set is a metric affine space isometric to the space of upper triangular matrices endowed with the flag rank metric. This suggests the development of a theory for flag rank metric codes in analogy to the rank metric codes used in linear subspace coding.</p>

An Efficient Post-Quantum Attribute-Based Encryption Scheme Based on Rank Metric Codes for Cloud Computing

An Efficient Post-Quantum Attribute-Based Encryption Scheme Based on Rank Metric Codes for Cloud Computing

Performance Improvement of Kötter and Kschischang Codes and Lifted Rank Metric Codes in Random Linear Network Coding

Performance Improvement of Kötter and Kschischang Codes and Lifted Rank Metric Codes in Random Linear Network Coding

A New Rank Metric Codes based Identification Scheme

A New Rank Metric Codes based Identification Scheme

The geometry of one-weight codes in the sum-rank metric

We provide a geometric characterization of k-dimensional Fqm-linear sum-rank metric codes as tuples of Fq-subspaces of Fqmk. We then use this characterization to study one-weight codes in the sum-rank metric. This leads us to extend the family of linearized Reed-Solomon codes in order to obtain a doubly-extended version of them. We prove that these codes are still maximum sum-rank distance (MSRD) codes and, when k=2, they are one-weight, as in the Hamming-metric case. We then focus on constant rank-profile codes in the sum-rank metric, which are a special family of one weight-codes, and derive constraints on their parameters with the aid of an associated Hamming-metric code. Furthermore, we introduce the n-simplex codes in the sum-rank metric, which are obtained as the orbit of a Singer subgroup of GL(k,qm). They turn out to be constant rank-profile – and hence one-weight – and generalize the simplex codes in both the Hamming and the rank metric. Finally, we focus on 2-dimensional one-weight codes, deriving constraints on the parameters of those which are also MSRD, and we find a new construction of one-weight MSRD codes when q=2.

Automated filtering scripts based on accurate mass applied to comprehensive two-dimensional gas chromatography coupled to high resolution mass spectrometry for group-type analysis of naphthenic acids

In this work, it is reported the detailed evaluation of filtering scripts for comprehensive two-dimensional gas chromatography coupled with high-resolution mass spectrometry (GC×GC–HRMS). Automated scripts were built to perform group-type analysis of naphthenic acids (NAs) in complex mixtures, like produced water (PW) samples. The logical operations were based on high-resolution mass spectral information acquired by using electron ionization (EI) and chemical ionization in the positive mode (PCI). The most common derivatization methods were carefully evaluated for group-type analysis. The performance of the scripts was evaluated using the rank metric (RM), wherein the sum-normalized absolute difference was calculated for true positives versus false positives. The best performing approach for qualitative analysis was attained by combining the derivatization with N-tert-butyldimethylsilyl-N-methyltrifluoroacetamide (MTBSTFA) with EI due to the high intensity and characteristic pseudo-molecular ions (even-electron [M-57]+). The data obtained supported the development of selective scripts, exhibiting accuracy values above 98 % and 88 % for the reference and unknown PW samples, respectively. The retention pattern of GC×GC was also used as a discriminant parameter for ambiguous identification, highlighting the synergy between GC×GC and HRMS. The proposed scripts are an interesting alternative to profile 154 classes of MTBSTFA-derived NAs, which considerably reduced the time spent in manual data curation and annotation, while improving the reliability of the results. This is an important step to aid in the development of standard methods for the study and regulation of water-soluble compounds found in PW samples.

Identification of the molecular subgroups in Alzheimer's disease by transcriptomic data.

BackgroundAlzheimer's disease (AD) is a heterogeneous pathological disease with genetic background accompanied by aging. This inconsistency is present among molecular subtypes, which has led to diagnostic ambiguity and failure in drug development. We precisely distinguished patients of AD at the transcriptome level.MethodsWe collected 1,240 AD brain tissue samples collected from the GEO dataset. Consensus clustering was used to identify molecular subtypes, and the clinical characteristics were focused on. To reveal transcriptome differences among subgroups, we certificated specific upregulated genes and annotated the biological function. According to RANK METRIC SCORE in GSEA, TOP10 was defined as the hub gene. In addition, the systematic correlation between the hub gene and “A/T/N” was analyzed. Finally, we used external data sets to verify the diagnostic value of hub genes.ResultsWe identified three molecular subtypes of AD from 743 AD samples, among which subtypes I and III had high-risk factors, and subtype II had protective factors. All three subgroups had higher neuritis plaque density, and subgroups I and III had higher clinical dementia scores and neurofibrillary tangles than subgroup II. Our results confirmed a positive association between neurofibrillary tangles and dementia, but not neuritis plaques. Subgroup I genes clustered in viral infection, hypoxia injury, and angiogenesis. Subgroup II showed heterogeneity in synaptic pathology, and we found several essential beneficial synaptic proteins. Due to presenilin one amplification, Subgroup III was a risk subgroup suspected of familial AD, involving abnormal neurogenic signals, glial cell differentiation, and proliferation. Among the three subgroups, the highest combined diagnostic value of the hub genes were 0.95, 0.92, and 0.83, respectively, indicating that the hub genes had sound typing and diagnostic ability.ConclusionThe transcriptome classification of AD cases played out the pathological heterogeneity of different subgroups. It throws daylight on the personalized diagnosis and treatment of AD.

Ouroboros: An Efficient and Provably Secure KEM Family

In this paper we introduce Ouroboros, a new family of Key Exchange protocols based on coding theory. The protocols propose a middle ground between the cryptosystems based on <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathsf {QC}$ </tex-math></inline-formula> - <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathsf {MDPC}$ </tex-math></inline-formula> codes, which feature small parameter sizes, but have a security reduction to two problems: the syndrome decoding problem and the indistinguishability of the code, and the HQC protocol, which features bigger parameters but has a security reduction to the syndrome decoding problem only. Ouroboros features a reduction to the syndrome decoding problem with only a small overhead compared to the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathsf {QC}$ </tex-math></inline-formula> - <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathsf {MDPC}$ </tex-math></inline-formula> based cryptosystems. The approach is based on an ideal structure and also works for the rank metric. This yields a simple, secure and efficient approach for key exchange, the Ouroboros family of protocols. For the Hamming metric we obtain the same type of parameters (and almost the same simple decoding) as for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathsf {MDPC}$ </tex-math></inline-formula> based cryptosystems, but with a security reduction to decoding random quasi-cyclic codes in the Random Oracle Model. This represents a reduction of up to 38% on the public key size compared to HQC, for the most secure parameters. For the rank metric, we obtain better parameters than for RQC, saving up to 31% on the public key for the most secure set of parameters, using non homogeneous errors in Ouroboros. In this full version, the protocol and decoding algorithm have been slightly improved, additional details are given in the security proof, and the protocol is fully described for the rank metric.

Generic Decoding in the Sum-Rank Metric

We propose the first non-trivial generic decoding algorithm for codes in the sum-rank metric. The new method combines ideas of well-known generic decoders in the Hamming and rank metric. For the same code parameters and number of errors, the new generic decoder has a larger expected complexity than the known generic decoders for the Hamming metric and smaller than the known rank-metric decoders. Furthermore, we give a formal hardness reduction, providing evidence that generic sum-rank decoding is computationally hard. As a by-product of the above, we solve some fundamental coding problems in the sum-rank metric: we give an algorithm to compute the exact size of a sphere of a given sum-rank radius, and also give an upper bound as a closed formula; and we study erasure decoding with respect to two different notions of support.

Optimal Anticodes, MSRD Codes, and Generalized Weights in the Sum-Rank Metric

Sum-rank metric codes have recently attracted the attention of many researchers, due to their relevance in several applications. Mathematically, the sum-rank metric is a natural generalization of both the Hamming metric and the rank metric. In this paper, we provide an Anticode Bound for the sum-rank metric, which extends the corresponding Hamming and rank-metric Anticode bounds. We classify then optimal anticodes, i.e., codes attaining the sum-rank metric Anticode Bound. We use these optimal anticodes to define generalized sum-rank weights and we study their main properties. In particular, we prove that the generalized weights of an MSRD code are determined by its parameters. As an application, in the Appendix we explain how generalized weights measure information leakage in multishot network coding.

Common Complements of Linear Subspaces and the Sparseness of MRD Codes

We consider the problem of estimating the number of common complements of a family of subspaces over a finite field, in terms of the cardinality of the family and its intersection structure. We derive upper and lower bounds for this number, along with their asymptotic versions as the field size tends to infinity. We use these bounds to describe the general behavior of common complements with respect to sparsity and density, showing that the decisive property is whether or not the number of spaces to be complemented is negligible with respect to the field size. The proof techniques are based on the study of isolated vertices in certain bipartite graphs. By specializing our results to matrix spaces, we answer an open question in coding theory, proving that MRD codes in the rank metric are sparse for all parameter sets as the field grows, with only very few exceptions. We also investigate the density of MRD codes as their column length tends to infinity, obtaining a new asymptotic bound. Using properties of the Euler function from number theory, we then show that our bound improves on known results for most parameter sets. We conclude the paper by establishing two structural properties of the density function of rank-metric codes.