/pdf/rank-metric-codes-and-their-applications-1cilo4zr.pdf

Rank-Metric Codes and Their Applications

/pdf/anticodes-in-the-sum-rank-metric-1j0qsaqo.pdf

Anticodes in the sum-rank metric

We study the structure of anticodes in the sum-rank metric for arbitrary fields and matrix blocks of arbitrary sizes. Our main result is a complete classification of optimal linear anticodes. We also compare the cardinality of the ball in the sum-rank metric with that of an optimal linear anticode, showing that the latter is strictly larger over sufficiently large finite fields. Finally, we give examples of parameters for which the largest anticode is neither a ball nor a linear anticode

Anticodes in the Sum-Rank Metric.

Rank-metric codes date back to the 1970s and today play a vital role in many areas of coding theory and cryptography. In this survey the authors provide a comprehensive overview of the known properties of rank-metric codes and their applications. The authors begin with an accessible and complete introduction to rank-metric codes, their properties and their decoding. They then discuss at length rank-metric code-based quantum resistant encryption and authentication schemes. The application of rank-metric codes to distributed data storage is also outlined. Finally, the constructions of network codes based on MRD codes, constructions of subspace codes by lifting rank-metric codes, bounds on the cardinality, and the list decoding capability of subspace codes is covered in depth. Rank-Metric Codes and Their Applications provides the reader with a concise, yet complete, general introduction to rank-metric codes, explains their most important applications, and highlights their relevance to these areas of research. 

Abstract 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. 

https://link.springer.com/content/pdf/10.1007/s40863-022-00314-y.pdf

Weight spectra of Gabidulin rank-metric codes and Betti numbers

Rank-Metric Codes, Generalized Binomial Moments and their Zeta Functions

In this work we introduce a new class of optimal tensor codes related to the Ravagnani-type anticodes, namely the j -tensor maximum rank distance codes. We show that it extends the family of j -maximum rank distance codes and contains the j -tensor binomial moment determined codes (with respect to the Ravagnani-type anticodes) as a proper subclass. We deﬁne and study the generalized zeta function for tensor codes. We establish connections between this object and the weight enumerator of a code with respect to the Ravagnani-type anticodes. We introduce a new reﬁnement of the invariants of tensor codes exploiting the structure of product lattices of some classes of anticodes and we derive the corresponding MacWilliams identities. In this framework, we also deﬁne a multivariate version of the tensor weight enumerator and we establish relations with the corresponding zeta function. As an application we derive connections on the generalized tensor weights related to the Delsarte and Ravagnani-type anticodes.

Zeta Functions for Tensor Codes

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 one 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 one 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 $\mathbb{F}_{q^m}$-linear codes and we show that they are extremal with respect to Kruskal's tensor rank bound.

Bilinear Complexity of 3-Tensors Linked to Coding Theory.

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.

Rank-Metric Lattices

We investigate adversarial network coding and decoding focusing on the multishot regime. Errors can occur on a proper subset of the network edges and are modeled via an adversarial channel. The paper contains both bounds and capacity-achieving schemes for the Diamond Network and the Mirrored Diamond Network. We also initiate the study of the generalizations of these networks.

Multishot Adversarial Network Decoding

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