Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userEfficient Decoding of Gabidulin Codes over Galois Rings
136
0
0.0
(
0
)
Added by
Julian Renner
Publication date
2021
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
This paper presents the first decoding algorithm for Gabidulin codes over Galois rings with provable quadratic complexity. The new method consists of two steps: (1) solving a syndrome-based key equation to obtain the annihilator polynomial of the error and therefore the column space of the error, (2) solving a key equation based on the received word in order to reconstruct the error vector. This two-step approach became necessary since standard solutions as the Euclidean algorithm do not properly work over rings.
rate research
Read More
Given $texttt{S}|texttt{R}$ a finite Galois extension of finite chain rings and $mathcal{B}$ an $texttt{S}$-linear code we define two Galois operators, the closure operator and the interior operator. We proof that a linear code is Galois invariant if and only if the row standard form of its generator matrix has all entries in the fixed ring by the Galois group and show a Galois correspondence in the class of $texttt{S}$-linear codes. As applications some improvements of upper and lower bounds for the rank of the restriction and trace code are given and some applications to $texttt{S}$-linear cyclic codes are shown.
We address the problem of decoding Gabidulin codes beyond their unique error-correction radius. The complexity of this problem is of importance to assess the security of some rank-metric code-based cryptosystems. We propose an approach that introduces row or column erasures to decrease the rank of the error in order to use any proper polynomial-time Gabidulin code error-erasure decoding algorithm. This approach improves on generic rank-metric decoders by an exponential factor.
In this paper, we determine the covering radius and a class of deep holes for Gabidulin codes with both rank metric and Hamming metric. Moreover, we give a necessary and sufficient condition for deciding whether a word is not a deep hole for Gabidulin codes, by which we study the error distance of a special class of words to certain Gabidulin codes.
A structure theorem of the group codes which are relative projective for the subgroup $lbrace 1 rbrace$ of $G$ is given. With this, we show that all such relative projective group codes in a fixed group algebra $RG$ are in bijection to the chains of projective group codes of length $ell$ in the group algebra $mathbb{F}G$, where $mathbb{F}$ is the residue field of $R$. We use a given chain to construct the dual code in $RG$ and also derive the minimum Hamming weight as well as a lower bound of the minimum euclidean weight.
In this paper we give the generalization of lifted codes over any finite chain ring. This has been done by using the construction of finite chain rings from $p$-adic fields. Further we propose a lattice construction from linear codes over finite chain rings using lifted codes.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
