Related papers: Improved decoding of symmetric rank metric errors
Codes in the sum-rank metric have various applications in error control for multishot network coding, distributed storage and code-based cryptography. Linearized Reed-Solomon (LRS) codes contain Reed-Solomon and Gabidulin codes as…
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…
In this paper, we give a generalization on the error correcting capability of twisted centralizer codes obtained from a fixed rank 1 matrix. In particular, we fix the combinatorial matrix which is obtained by getting the linear combination…
In the present paper, we consider list decoding for both random rank metric codes and random linear rank metric codes. Firstly, we show that, for arbitrary $0<R<1$ and $\epsilon>0$ ($\epsilon$ and $R$ are independent), if $0<\frac{n}{m}\leq…
Random network coding recently attracts attention as a technique to disseminate information in a network. This paper considers a non-coherent multi-shot network, where the unknown and time-variant network is used several times. In order to…
We show that the sequence of dimensions of the linear spaces, generated by a given rank-metric code together with itself under several applications of a field automorphism, is an invariant for the whole equivalence class of the code. These…
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…
We introduce a new family of rank metric codes: Low Rank Parity Check codes (LRPC), for which we propose an efficient probabilistic decoding algorithm. This family of codes can be seen as the equivalent of classical LDPC codes for the rank…
This paper considers the problem of channel coding with a given (possibly suboptimal) maximum-metric decoding rule. A cost-constrained random-coding ensemble with multiple auxiliary costs is introduced, and is shown to achieve error…
Recursive decoding techniques are considered for Reed-Muller (RM) codes of growing length $n$ and fixed order $r.$ An algorithm is designed that has complexity of order $n\log n$ and corrects most error patterns of weight up to…
This paper studies random-coding error exponents of randomised list decoding, in which the decoder randomly selects $L$ messages with probabilities proportional to the decoding metric of the codewords. The exponents (or bounds) are given…
We present error-correcting codes that achieve the information-theoretically best possible trade-off between the rate and error-correction radius. Specifically, for every $0 < R < 1$ and $\eps> 0$, we present an explicit construction of…
In this paper we introduce and investigate rank-metric intersecting codes, a new class of linear codes in the rank-metric context, inspired by the well-studied notion of intersecting codes in the Hamming metric. A rank-metric code is said…
The list-decodability of random linear rank-metric codes is shown to match that of random rank-metric codes. Specifically, an $\mathbb{F}_q$-linear rank-metric code over $\mathbb{F}_q^{m \times n}$ of rate $R =…
Gabidulin codes, originally defined over finite fields, are an important class of rank metric codes with various applications. Recently, their definition was generalized to certain fields of characteristic zero and a Welch--Berlekamp like…
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…
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,…
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…
In this paper, we present an optimal metric function on average, which leads to a significantly low decoding computation while maintaining the superiority of the polarization-adjusted convolutional (PAC) codes' error-correction performance.…
Linear codes correcting one deletions have rate at most $1/2$. In this paper, we construct linear list decodable codes correcting edits with rate approaching $1$ and reasonable list size. Our encoder and decoder run in polynomial time.