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.

scholarly journals Projection Decoding of Some Binary Optimal Linear Codes of Lengths 36 and 40

Mathematics ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 15
Author(s):  
Lucky Galvez ◽  
Jon-Lark Kim
Keyword(s):  
Linear Codes ◽  
Cyclic Codes ◽  
Minimum Weight ◽  
Error Correcting Codes ◽  
Decoding Algorithm ◽  
Decoding Algorithms ◽  
Syndrome Decoding ◽  
Optimal Linear ◽  
Optimal Linear Codes ◽  
Efficient Decoding

Practically good error-correcting codes should have good parameters and efficient decoding algorithms. Some algebraically defined good codes, such as cyclic codes, Reed–Solomon codes, and Reed–Muller codes, have nice decoding algorithms. However, many optimal linear codes do not have an efficient decoding algorithm except for the general syndrome decoding which requires a lot of memory. Therefore, a natural question to ask is which optimal linear codes have an efficient decoding. We show that two binary optimal [ 36 , 19 , 8 ] linear codes and two binary optimal [ 40 , 22 , 8 ] codes have an efficient decoding algorithm. There was no known efficient decoding algorithm for the binary optimal [ 36 , 19 , 8 ] and [ 40 , 22 , 8 ] codes. We project them onto the much shorter length linear [ 9 , 5 , 4 ] and [ 10 , 6 , 4 ] codes over G F ( 4 ) , respectively. This decoding algorithm, called projection decoding, can correct errors of weight up to 3. These [ 36 , 19 , 8 ] and [ 40 , 22 , 8 ] codes respectively have more codewords than any optimal self-dual [ 36 , 18 , 8 ] and [ 40 , 20 , 8 ] codes for given length and minimum weight, implying that these codes are more practical.

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

scholarly journals The Decoding Algorithms For Error-Correcting Product Codes Based On Project Geometry Low-Density Parity-Check Codes

2020 ◽  
Vol 12 (3) ◽  
pp. 399-406
Author(s):  
Lev E. Nazarov ◽  
Keyword(s):  
Turbo Codes ◽  
Ldpc Codes ◽  
Parity Check ◽  
Decoding Algorithm ◽  
Signal Noise ◽  
Product Codes ◽  
Decoding Algorithms ◽  
Low Density Parity Check ◽  
Code Constructions ◽  
Parity Check Codes

The focus of this paper is directed towards the investigation of the characteristics of symbol-by-symbol iterative decoding algorithms for error-correcting block product-codes (block turbo-codes) which enable to reliable information transfer at relatively low received signal/noise and provide high power efficiency. Specific feature of investigated product codes is construction with usage of low-density parity-check codes (LDPC) and these code constructions are in the class of LDPC too. According to this fact the considered code constructions have symbol-by-symbol decoding algorithms developed for total class LDPC codes, namely BP (belief propagation) and its modification MIN_SUM_BP. The BP decoding algorithm is iterative and for implementation the signal/noise is required, for implementation of MIN_SUM_BP decoding algorithm the signal/noise is not required. The resulted characteristics of product codes constructed with usage of LDPC based on project geometry (length of code words, information volume, code rate, error performances) are presented in this paper. These component LDPC codes are cyclic and have encoding and decoding algorithms with low complexity implementation. The computer simulations for encoding and iterative symbol-by-symbol decoding algorithms for the number of turbo-codes with different code rate and information volumes are performed. The results of computer simulations have shown that MIN_SUM_BP decoding algorithm is more effective than BP decoding algorithm for channel with additive white gaussian noise concerning error-performances.

scholarly journals Determinants of some special matrices over commutative finite chain rings

2020 ◽  
Vol 8 (1) ◽  
pp. 242-256
Author(s):  
Somphong Jitman
Keyword(s):  
Finite Fields ◽  
Principal Ideal ◽  
Residue Field ◽  
Circulant Matrices ◽  
Finite Chain ◽  
Algebraic Structures ◽  
Open Problems ◽  
Finite Chain Rings ◽  
Positive Integers ◽  
Finite Principal Ideal Rings

AbstractCirculant matrices over finite fields and over commutative finite chain rings have been of interest due to their nice algebraic structures and wide applications. In many cases, such matrices over rings have a closed connection with diagonal matrices over their extension rings. In this paper, the determinants of diagonal and circulant matrices over commutative finite chain rings R with residue field 𝔽q are studied. The number of n × n diagonal matrices over R of determinant a is determined for all elements a in R and for all positive integers n. Subsequently, the enumeration of nonsingular n × n circulant matrices over R of determinant a is given for all units a in R and all positive integers n such that gcd(n, q) = 1. In some cases, the number of singular n × n circulant matrices over R with a fixed determinant is determined through the link between the rings of circulant matrices and diagonal matrices. As applications, a brief discussion on the determinants of diagonal and circulant matrices over commutative finite principal ideal rings is given. Finally, some open problems and conjectures are posted

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