scholarly journals Linear Codes over Finite Chain Rings

10.37236/1489 ◽  
1999 ◽  
Vol 7 (1) ◽  
Author(s):  
Thomas Honold ◽  
Ivan Landjev
Keyword(s):  
Linear Codes ◽  
Finite Chain ◽  
Finite Chain Rings ◽  
One To One

The aim of this paper is to develop a theory of linear codes over finite chain rings from a geometric viewpoint. Generalizing a well-known result for linear codes over fields, we prove that there exists a one-to-one correspondence between so-called fat linear codes over chain rings and multisets of points in projective Hjelmslev geometries, in the sense that semilinearly isomorphic codes correspond to equivalent multisets and vice versa. Using a selected class of multisets we show that certain MacDonald codes are linearly representable over nontrivial chain rings.

scholarly journals Decoding Linear Codes over Chain Rings Given by Parity Check Matrices

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1878
Author(s):  
José Gómez-Torrecillas ◽  
F. J. Lobillo ◽  
Gabriel Navarro
Keyword(s):  
Linear Codes ◽  
Residue Field ◽  
Finite Chain ◽  
Parity Check ◽  
Decoding Algorithm ◽  
Decoding Algorithms ◽  
Finite Chain Rings

We design a decoding algorithm for linear codes over finite chain rings given by their parity check matrices. It is assumed that decoding algorithms over the residue field are known at each degree of the adic decomposition.

A class of linear codes of length 2 over finite chain rings

2019 ◽  
Vol 19 (06) ◽  
pp. 2050103 ◽  
Author(s):  
Yonglin Cao ◽  
Yuan Cao ◽  
Hai Q. Dinh ◽  
Fang-Wei Fu ◽  
Jian Gao ◽  
...  
Keyword(s):  
Positive Integer ◽  
Linear Codes ◽  
Invertible Element ◽  
Irreducible Polynomial ◽  
Constacyclic Codes ◽  
Finite Chain ◽  
Chain Ring ◽  
Finite Chain Ring ◽  
Finite Chain Rings ◽  
Positive Integers

Let [Formula: see text] be a finite field of cardinality [Formula: see text], where [Formula: see text] is an odd prime, [Formula: see text] be positive integers satisfying [Formula: see text], and denote [Formula: see text], where [Formula: see text] is an irreducible polynomial in [Formula: see text]. In this note, for any fixed invertible element [Formula: see text], we present all distinct linear codes [Formula: see text] over [Formula: see text] of length [Formula: see text] satisfying the condition: [Formula: see text] for all [Formula: see text]. This conclusion can be used to determine the structure of [Formula: see text]-constacyclic codes over the finite chain ring [Formula: see text] of length [Formula: see text] for any positive integer [Formula: see text] satisfying [Formula: see text].

scholarly journals Galois correspondence on linear codes over finite chain rings

2020 ◽  
Vol 343 (2) ◽  
pp. 111653
Author(s):  
Alexandre Fotue Tabue ◽  
Edgar Martínez-Moro ◽  
Christophe Mouaha
Keyword(s):  
Linear Codes ◽  
Finite Chain ◽  
Finite Chain Rings ◽  
Galois Correspondence

scholarly journals Gilbert-Varshamov type bounds for linear codes over finite chain rings

2007 ◽  
Vol 1 (1) ◽  
pp. 99-109 ◽  
Author(s):  
Ferruh Özbudak ◽  
◽  
Patrick Solé ◽  
Keyword(s):  
Linear Codes ◽  
Finite Chain ◽  
Finite Chain Rings

Critical Problem for codes over finite chain rings

2021 ◽  
Vol 76 ◽  
pp. 101900
Author(s):  
Koji Imamura ◽  
Keisuke Shiromoto
Keyword(s):  
Finite Chain ◽  
Critical Problem ◽  
Finite Chain Rings
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