相关论文: On the hardness of code equivalence problems in ra…

Rank-metric codes were studied by E. Gabidulin in 1985 after a brief introduction by Delsarte in 1978 as an equivalent of Reed-Solomon codes, but based on linearized polynomials. They have found applications in many areas, including linear…

We consider the problem of distinguishing two vectors (visualized as images or barcodes) and learning if they are related to one another. For this, we develop a geometric quantum machine learning (GQML) approach with embedded symmetries…

In rank-metric cryptography, a vector from a finite dimensional linear space over a finite field is viewed as the linear space spanned by its entries. The rank decoding problem which is the analogue of the problem of decoding a random…

The graph matching optimization problem is an essential component for many tasks in computer vision, such as bringing two deformable objects in correspondence. Naturally, a wide range of applicable algorithms have been proposed in the last…

The nonnegative integer rank of a matrix is a variant of the classical nonnegative rank, introduced in the 1980s, where factorizations are required to have integer entries. While computing nonnegative integer rank is generally very hard, we…

Let $\mathrm{Sym}_q(m)$ be the space of symmetric matrices in $\mathbb{F}_q^{m\times m}$. A subspace of $\mathrm{Sym}_q(m)$ equipped with the rank distance is called a symmetric rank-metric code. In this paper we study the covering…

The paper deals with the problem of deciding if two finite-dimensional linear subspaces over an arbitrary field are identical up to a permutation of the coordinates. This problem is referred to as the permutation code equivalence. We show…

Inspired by the work of Zhou "On equivalence of maximum additive symmetric rank-distance codes" (2020) based on the paper of Schmidt "Symmetric bilinear forms over finite fields with applications to coding theory" (2015), we investigate the…

In this article, we consider the decoding problem of affine Grassmann codes over nonbinary fields. We use matrices of different ranks to construct a large set consisting of parity checks of affine Grassmann codes, which are orthogonal with…

Rank-metric codes, defined as sets of matrices over a finite field with the rank distance, have gained significant attention due to their applications in network coding and connections to diverse mathematical areas. Initially studied by…

Let $H\_0, ..., H\_n$ be $m \times m$ matrices with entries in $\QQ$ and Hankel structure, i.e. constant skew diagonals. We consider the linear Hankel matrix $H(\vecx)=H\_0+\X\_1H\_1+...+\X\_nH\_n$ and the problem of computing sample points…

Rank-metric codes have been a central topic in coding theory due to their theoretical and practical significance, with applications in network coding, distributed storage, crisscross error correction, and post-quantum cryptography. Recent…

Huffman coding finds a prefix code that minimizes mean codeword length for a given probability distribution over a finite number of items. Campbell generalized the Huffman problem to a family of problems in which the goal is to minimize not…

Rank weights and generalized rank weights have been proven to characterize error and erasure correction, and information leakage in linear network coding, in the same way as Hamming weights and generalized Hamming weights describe classical…

The sum-rank metric provides a unifying framework that generalizes both the celebrated Hamming and rank metrics, and has found applications in areas such as network coding, distributed storage, and space-time coding. A central problem is to…

We study skew polycyclic codes over a finite field $\mathbb{F}_q$, associated with a skew polynomial $f(x) \in \mathbb{F}_q[x;\sigma]$, where $\sigma$ is an automorphism of $\mathbb{F}_q$. We start by proving the Roos-like bound for both…

We show that the network coding and index coding problems are equivalent. This equivalence holds in the general setting which includes linear and non-linear codes. Specifically, we present an efficient reduction that maps a network coding…

Low Rank Parity Check (LRPC) codes form a class of rank-metric error-correcting codes that was purposely introduced to design public-key encryption schemes. An LRPC code is defined from a parity check matrix whose entries belong to a…

The study of linear codes over a finite field of odd cardinality, derived from determinantal varieties obtained from symmetric matrices of bounded rank, was initiated in a recent paper by the authors. There, one found the minimum distance…

This paper investigates the relationship between the rank weight distribution of a linear code and that of its dual code. The main result of this paper is that, similar to the MacWilliams identity for the Hamming metric, the rank weight…