Related papers: Sum-rank metric codes
Let $\mathbb{F}_q$ denote the finite field with $q=p^\lambda$ elements. Maximum Rank-metric codes (MRD for short) are subsets of $M_{m\times n}(\mathbb{F}_q)$ whose number of elements attains the Singleton-like bound. The first MRD codes…
This paper investigates packing and covering properties of codes with the rank metric. First, we investigate packing properties of rank metric codes. Then, we study sphere covering properties of rank metric codes, derive bounds on their…
Linearized Reed-Solomon (LRS) codes are sum-rank metric codes that fulfill the Singleton bound with equality. In the two extreme cases of the sum-rank metric, they coincide with Reed-Solomon codes (Hamming metric) and Gabidulin codes (rank…
MDS codes have diverse practical applications in communication systems, data storage, and quantum codes due to their algebraic properties and optimal error-correcting capability. In this paper, we focus on a class of linear codes and…
Rank modulation has been recently proposed as a scheme for storing information in flash memories. While rank modulation has advantages in improving write speed and endurance, the current encoding approach is based on the "push to the top"…
Reed-Muller (RM) codes are among the oldest, simplest and perhaps most ubiquitous family of codes. They are used in many areas of coding theory in both electrical engineering and computer science. Yet, many of their important properties are…
We study the symmetric tensor rank of multiplication over finite field extensions using linearized polynomials. Via field trace, symmetric linearized polynomials are identified with symmetric bilinear forms and symmetric matrices, allowing…
Universal security over a network with linear network coding has been intensively studied. However, previous linear codes and code pairs used for this purpose were linear over a larger field than that used on the network, which restricts…
We propose a unified theory of generalized weights for linear codes endowed with an arbitrary distance. Instead of relying on supports or anticodes, the weights of a code are defined via the intersections of the code with a chosen family of…
We consider the local rank-modulation scheme in which a sliding window going over a sequence of real-valued variables induces a sequence of permutations. Local rank-modulation is a generalization of the rank-modulation scheme, which has…
We define almost affine vector rank-metric codes as subsets $\mathcal{C}\subseteq \mathbb{F}_{q^m}^n$ whose canonical projections have cardinalities that are powers of $q^m$, and prove that they naturally induce $q$-matroids. We establish…
In this letter, we evaluate the finite-length performance of network coding when using either random or structured encoding matrices. First, we present our novel construction of structured network codes over Fq (q = 2^m) using Pascal…
We study information leakage in secure linear network coding schemes based on nested rank-metric codes. We show that the amount of information leaked to an adversary that observes a subset of network links is characterized by the…
The rank-modulation scheme has been recently proposed for efficiently storing data in nonvolatile memories. Error-correcting codes are essential for rank modulation, however, existing results have been limited. In this work we explore a new…
A multifold $1$-perfect code ($1$-perfect code for list decoding) in any graph is a set $C$ of vertices such that every vertex of the graph is at distance not more than $1$ from exactly $\mu$ elements of $C$. In $q$-ary Hamming graphs,…
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…
Rank modulation is a way of encoding information to correct errors in flash memory devices as well as impulse noise in transmission lines. Modeling rank modulation involves construction of packings of the space of permutations equipped with…
Low rank approximation is an important tool used in many applications of signal processing and machine learning. Recently, randomized sketching algorithms were proposed to effectively construct low rank approximations and obtain approximate…
A class of linear block codes which simultaneously generalizes Gabidulin codes and a class of skew cyclic codes is defined. For these codes, both a Hartmann-Tzeng-like bound and a Roos-like bound, with respect to their rank distance, are…
An index coding (IC) problem consisting of a server and multiple receivers with different side-information and demand sets can be equivalently represented using a fitting matrix. A scalar linear index code to a given IC problem is a matrix…