Related papers: Linear rank-metric intersecting codes
For a growing number of applications such as cellular, peer-to-peer, and sensor networks, efficient error-free transmission of data through a network is essential. Toward this end, K\"{o}tter and Kschischang propose the use of subspace…
The sum-rank metric is the mixture of the Hamming and rank metrics. The sum-rank metric found its application in network coding, locally repairable codes, space-time coding, and quantum-resistant cryptography. Linearized Reed-Solomon (LRS)…
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,…
Let $F$ be a field and let $F^{r\times s}$ denote the space of $r\times s$ matrices over $F$. Given equinumerous subsets $\mathcal{A}=\{A_i\mid i \in I\}\subseteq F^{r\times r}$ and $\mathcal{B}=\{B_i\mid i\in I\}\subseteq F^{s\times s}$ we…
The projective space $\mathbb{P}_q(n)$, i.e. the set of all subspaces of the vector space $\mathbb{F}_q^n$, is a metric space endowed with the subspace distance metric. Braun, Etzion and Vardy argued that codes in a projective space are…
A new construction is proposed for low density parity check (LDPC) codes using quadratic permutation polynomials over finite integer rings. The associated graphs for the new codes have both algebraic and pseudo-random nature, and the new…
We give a complete characterization of simple graphs whose adjacency matrices generate binary linear complementary dual (LCD) codes. In particular, we completely characterize a distance-regular graph which yields an LCD code in terms of the…
This paper investigates the problem of single-source multicasting over a communication network in the presence of restricted adversaries. When the adversary is constrained to operate only on a prescribed subset of edges, classical cut-set…
The projective space of order $n$ over a finite field $\F_q$ is a set of all subspaces of the vector space $\F_q^{n}$. In this work, we consider error-correcting codes in the projective space, focusing mainly on constant dimension codes. We…
In 2021, Augot, Couvreur, Lavauzelle and Neri introduced a new class of rank metric codes which can be regarded as rank metric counterparts of Reed-Muller codes. Given a finite Galois extension $\mathbb{L} / \mathbb{K}$, these codes are…
Coding in the projective space has received recently a lot of attention due to its application in network coding. Reduced row echelon form of the linear subspaces and Ferrers diagram can play a key role for solving coding problems in the…
This paper examines linear binary codes capable of correcting one or more errors. For the single-error-correcting case, it is shown that the Hamming bound is achieved by a constructive method, and an exact expression for the minimal…
We propose and analyze an interleaved variant of Loidreau's rank-metric cryptosystem based on rank multipliers. We analyze and adapt several attacks on the system, propose design rules, and study weak keys. Finding secure instances requires…
We examine an error-correcting coding framework in which each coded symbol is constrained to be a function of a fixed subset of the message symbols. With an eye toward distributed storage applications, we seek to design systematic codes…
Determining information ratios of access structures is an important problem in secret sharing. Information inequalities and linear rank inequalities play an important role for proving bounds. Characteristic-dependent linear rank…
A subspace of matrices over $\mathbb{F}_{q^e}^{m\times n}$ can be naturally embedded as a subspace of matrices in $\mathbb{F}_q^{em\times en}$ with the property that the rank of any of its matrix is a multiple of $e$. It is quite natural to…
A classical method of constructing a linear code over $\gf(q)$ with a $t$-design is to use the incidence matrix of the $t$-design as a generator matrix over $\gf(q)$ of the code. This approach has been extensively investigated in the…
We propose a new class of space-time block codes based on finite-field rank-metric codes in combination with a rank-metric-preserving mapping to the set of Eisenstein integers. It is shown that these codes achieve maximum diversity order…
We combine standard persistent homology with image persistent homology to define a novel way of characterizing shapes and interactions between them. In particular, we introduce: (1) a mixup barcode, which captures geometric-topological…
In this paper, we study the minimum distances of binary linear codes with parity check matrices formed from subset inclusion matrices $W_{t,n,k}$, representing $t$-element subsets versus $k$-element subsets of an $n$-element set. We provide…