Rank-metric Codes Research Articles

Short Rank-Metric Codes and Scattered Subspaces

Short Rank-Metric Codes and Scattered Subspaces

New constant dimension subspace codes from multilevel linkage construction

<p style='text-indent:20px;'>One of the main problems in subspace coding is to determine the maximal possible size of a constant dimension subspace code with given parameters. In this paper, we introduce a family of new codes, named rank-restricted Ferrers diagram rank-metric codes, to give an improved construction of constant dimension subspace codes. Our method constructs many new CDCs with larger sizes than previously best-known codes.</p>

On the rank decoding problem over finite principal ideal rings

The rank decoding problem has been the subject of much attention in this last decade. This problem, which is at the base of the security of public-key cryptosystems based on rank metric codes, is traditionally studied over finite fields. But the recent generalizations of certain classes of rank-metric codes from finite fields to finite rings have naturally created the interest to tackle the rank decoding problem in the case of finite rings.In this paper, we show that solving the rank decoding problem over finite principal ideal rings is at least as hard as the rank decoding problem over finite fields. We also show that computing the minimum rank distance for linear codes over finite principal ideal rings is equivalent to the same problem for linear codes over finite fields. Finally, we provide combinatorial type algorithms for solving the rank decoding problem over finite chain rings together with their average complexities.

Constructions of perfect bases for classes of 3-tensors

A well studied problem in algebraic complexity theory is the determination of the complexity of problems relying on evaluations of bilinear maps. One measure of the complexity of a bilinear map (or 3-tensor) is the optimal number of non-scalar multiplications required to evaluate it. This quantity is also described as its tensor rank, which is the smallest number of rank-1 matrices whose span contains its first slice space. In this paper we derive upper bounds on the tensor ranks of certain classes of 3-tensors and give explicit constructions of sets of rank-1 matrices containing their first slice spaces. We also show how these results can be applied in coding theory to derive upper bounds on the tensor rank of some rank-metric codes. In particular, we compute the tensor rank of some families of Fqm-linear codes and we show that they are extremal with respect to Kruskal's tensor rank bound.

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.

Insdel codes from subspace and rank-metric codes

Insertion and deletion (insdel in short) codes are designed to deal with synchronization errors in communication channels caused by insertions and deletions of message symbols. These codes have received a lot of attention due to their applications in diverse areas such as computational biology, DNA data storage, race-track memory error corrections, language processing, and synchronous digital communication networks. In the present work, we study constructions and limitations of insdel codes from rank metric and subspace codes. This paper studies and improves the idea of the work [4] by Hao Chen on the connection between insdel codes and subspace codes.We discuss why subspace code is a natural choice for constructing insdel codes and show that the interleaved Gabidulin codes can be used to construct nonlinear insdel codes approaching the Singleton bound. Then we show that the indexing scheme of transforming efficient Hamming metric codes to efficient insdel codes can be adapted for the class of rank metric codes. And that improves the base field size of the construction of insdel codes from lifted rank-metric codes. It is also shown that the size of the insdel code from a subspace code can be improved significantly than in the previously proposed construction. We give an algebraic condition for a linear Gabidulin rank metric code to be optimal insdel code adapting the condition proved for Reed-Solomon codes. Moreover, we give constructions of linear and nonlinear insdel codes from Sidon spaces.

MacWilliams' Extension Theorem for rank-metric codes

The MacWilliams' Extension Theorem is a classical result by Florence Jessie MacWilliams. It shows that every linear isometry between linear block-codes endowed with the Hamming distance can be extended to a linear isometry of the ambient space. Such an extension fails to exist in general for rank-metric codes, that is, one can easily find examples of linear isometries between rank-metric codes which cannot be extended to linear isometries of the ambient space. In this paper, we explore to what extent a MacWilliams' Extension Theorem may hold for rank-metric codes. We provide an extensive list of examples of obstructions to the existence of an extension, as well as a positive result.

Evasive subspaces, generalized rank weights and near MRD codes

We revisit and extend the connections between Fqm-linear rank-metric codes and evasive Fq-subspaces of Fqmk. We give a unifying framework in which we prove in an elementary way how the parameters of a rank-metric code are related to special geometric properties of the associated evasive subspace, with a particular focus on the generalized rank weights. In this way, we can also provide alternative and very short proofs of known results on scattered subspaces. We then use this simplified point of view in order to get a geometric characterization of near MRD codes and a clear bound on their maximal length. Finally we connect the theory of quasi-MRD codes with h-scattered subspaces of maximum dimension, extending to all the parameters sets the already known results on MRD codes.

Rank-Metric Codes, Semifields, and the Average Critical Problem

We investigate two fundamental questions intersecting coding theory and combinatorial geometry, with emphasis on their connections. These are the problem of computing the asymptotic density of MRD codes in the rank metric, and the Critical Problem for combinatorial geometries by Crapo and Rota. In the first part of the paper, we use methods from semifield theory to derive two lower bounds for the density function of full-rank, square MRD codes. The first bound is sharp when the matrix size is a prime number and the underlying field is sufficiently large, while the second bound applies to the binary field. We then take a new look at the Critical Problem for combinatorial geometries, approaching it from a qualitative, often asymptotic, viewpoint. We illustrate the connection between this very classical problem and that of computing the asymptotic density of MRD codes. Finally, in the third part of the paper we study the asymptotic density of some special families of codes in the rank metric, including the symmetric, alternating, and Hermitian ones. In particular, we show that the optimal codes in these three contexts are sparse.

LCP of rank metric codes and its an application

Linear complementary pair (LCP) of codes and rank metric codes over finite fields have been intensively investigated recently due to their applications in cryptography. In this paper, we first give the definition of LCP of rank-metric codes combined by the LCP of codes and rank-metric codes, and we establish the relationship between LCP of Delsarte codes and LCP of Gabidulin codes. We then present a criterion for Gabidulin codes (C,D) to be LCP. Using this criterion, we obtain two classes of LCP MRD of Gabidulin codes. Finally, we exhibit an interesting application of LCP MRD of Gabidulin codes in coding for the two-user binary adder channel.

Quasi optimal anticodes: structure and invariants

It is well-known that the dimension of optimal anticodes in the rank-metric is divisible by the maximum m between the number of rows and columns of the matrices. Moreover, for a fixed k divisible by m, optimal rank-metric anticodes are the codes with least maximum rank, among those of dimension k. In this paper, we study the family of rank-metric codes whose dimension is not divisible by m and whose maximum rank is the least possible for codes of that dimension, according to the Anticode bound. As these are not optimal anticodes, we call them quasi optimal anticodes (qOACs). In addition, we call dually qOAC a qOAC whose dual is also a qOAC. We describe explicitly the structure of dually qOACs and compute their weight distributions, generalized weights, and associated q-polymatroids.

Rank-Metric Lattices

We introduce the class of rank-metric geometric lattices and initiate the study of their structural properties. Rank-metric lattices can be seen as the $q$-analogues of higher-weight Dowling lattices, defined by Dowling himself in 1971. We fully characterize the supersolvable rank-metric lattices and compute their characteristic polynomials. We then concentrate on small rank-metric lattices whose characteristic polynomial we cannot compute, and provide a formula for them under a polynomiality assumption on their Whitney numbers of the first kind. The proof relies on computational results and on the theory of vector rank-metric codes, which we review in this paper from the perspective of rank-metric lattices. More precisely, we introduce the notion of lattice-rank weights of a rank-metric code and investigate their properties as combinatorial invariants and as code distinguishers for inequivalent codes.

Constructions of optimal rank-metric codes from automorphisms of rational function fields

<p style='text-indent:20px;'>We define a class of automorphisms of rational function fields of finite characteristic and employ these to construct different types of optimal linear rank-metric codes. The first construction is of generalized Gabidulin codes over rational function fields. Reducing these codes over finite fields, we obtain maximum rank distance (MRD) codes which are not equivalent to generalized twisted Gabidulin codes. We also construct optimal Ferrers diagram rank-metric codes which settles further a conjecture by Etzion and Silberstein.</p>

Engineering Practical Rank-Code-Based Cryptographic Schemes on Embedded Hardware. A Case Study on ROLLO

In this paper, we investigate the practical performance of rank-code based cryptography on FPGA platforms by presenting a case study on the quantum-safe KEM scheme based on LRPC codes called ROLLO, which was among NIST post-quantum cryptography standardization round-2 candidates. Specifically, we present an FPGA implementation of the encapsulation and decapsulation operations of the ROLLO KEM scheme with some variations to the original specification. The design is fully parameterized, using code-generation scripts to support a wide range of parameter choices for security levels specified in ROLLO. At the core of the ROLLO hardware, we presented a generic approach for hardware-based Gaussian elimination, which can process both non-singular and singular matrices. Previous works on hardware-based Gaussian elimination can only process non-singular ones. However, a plethora of cryptosystems, for instance, quantum-safe key encapsulation mechanisms based on rank-metric codes, ROLLO and RQC, which are among NIST post-quantum cryptography standardization round-2 candidates, require performing Gaussian elimination for random matrices regardless of the singularity. To the best of our knowledge, this work is the first hardware implementation for rank-code-based cryptographic schemes. The experimental results suggest rank-code-based schemes can be highly efficient.

Double Multilevel Constructions for Constant Dimension Codes

Abstract: Constant dimension codes (CDCs), as special subspace codes, have received a lot of attention due to their application in random network coding. This paper introduces a family of new codes, called Ferrers diagram rank-metric codes with given ranks (GFRMCs), to generalize the parallel construction and the parallel multilevel construction in [IEEE Trans. Inf. Theory, 66 (2020), 6884–6897]. The lower bounds for GFRMCs are derived from Ferrers diagram rank-metric codes (FDRMCs). Via GFRMCs, the inverse multilevel construction for CDCs is showed. Furthermore, the double multilevel construction, as an effective construction for CDCs, is presented by combining the inverse multilevel construction and the multilevel construction. Many CDCs with larger size than the previously best known codes are given.

Equivalence and automorphism groups of two families of maximum scattered linear sets

Linear set in projective spaces over finite fields plays central roles in the study of blocking sets, semifields, rank-metric codes and etc. A linear set with the largest possible cardinality and the maximum rank is called maximum scattered. Despite two decades of study, there are only a few number of known maximum scattered linear sets in projective lines, including the family constructed by Csajb\'ok, Marino, Polverino and Zanella 2018, and the family constructed by Csajb\'ok, Marino, Zullo 2018 (also Marino, Montanucci, and Zullo 2020). This paper aims to solve the equivalence problem of the linear sets in each of these families and to determine their automorphism groups.

Twisted linearized Reed-Solomon codes: A skew polynomial framework

We provide an algebraic description for sum-rank metric codes, as quotient space of a skew polynomial ring. This approach generalizes at the same time the skew group algebra setting for rank-metric codes and the polynomial setting for codes in the Hamming metric. This allows to construct twisted linearized Reed-Solomon codes, a new family of maximum sum-rank distance codes extending at the same time Sheekey's twisted Gabidulin codes in the rank metric and twisted Reed-Solomon codes in the Hamming metric. Furthermore, we provide an analogue in the sum-rank metric of Trombetti-Zhou construction, which also provides a family of maximum sum-rank distance codes. As a byproduct, in the extremal case of the Hamming metric, we obtain a new family of additive MDS codes over quadratic fields.

Linear cutting blocking sets and minimal codes in the rank metric

This work investigates the structure of rank-metric codes in connection with concepts from finite geometry, most notably the q-analogues of projective systems and blocking sets. We also illustrate how to associate a classical Hamming-metric code to a rank-metric one, in such a way that various rank-metric properties naturally translate into the homonymous Hamming-metric notions under this correspondence. The most interesting applications of our results lie in the theory of minimal rank-metric codes, which we introduce and study from several angles. Our main contributions are bounds for the parameters of a minimal rank-metric codes, a general existence result based on a combinatorial argument, and an explicit code construction for some parameter sets that uses the notion of a scattered linear set. Throughout the paper we also show and comment on curious analogies/divergences between the theories of error-correcting codes in the rank and in the Hamming metric.

New MRD codes from linear cutting blocking sets

Minimal rank-metric codes or, equivalently, linear cutting blocking sets are characterized in terms of the second generalized rank weight, via their connection with evasiveness properties of the associated q-system. Using this result, we provide the first construction of a family of mathbb F_{q^m}-linear MRD codes of length 2m that are not obtained as a direct sum of two smaller MRD codes. In addition, such a family has better parameters, since its codes possess generalized rank weights strictly larger than those of the previously known MRD codes. This shows that not all the MRD codes have the same generalized rank weights, in contrast to what happens in the Hamming metric setting.

Weight spectra of Gabidulin rank-metric codes and Betti numbers

We consider q-matroids and their associated classical matroids derived from Gabidulin rank-metric codes. We express the generalized rank weights of a Gabidulin rank-metric code in terms of Betti numbers of the dual classical matroid associated to the q-matroid corresponding to the code. In our main result, we show how these Betti numbers and their elongations determine the generalized weight polynomials for q-matroids, in particular, for the Gabidulin rank-metric codes. In addition, we demonstrate how the weight distribution and higher weight spectra of such codes can be determined directly from the associated q-matroids by using Möbius functions of its lattice of q-flats.