scholarly journals Skew Constacyclic Codes over Finite Fields and Finite Chain Rings

2016 ◽  
Vol 2016 ◽  
pp. 1-17 ◽  
Author(s):  
Hai Q. Dinh ◽  
Bac T. Nguyen ◽  
Songsak Sriboonchitta
Keyword(s):  
Finite Fields ◽  
Bch Codes ◽  
Constacyclic Codes ◽  
Finite Chain ◽  
Dual Codes ◽  
Negacyclic Codes ◽  
Finite Chain Rings

This paper overviews the study of skewΘ-λ-constacyclic codes over finite fields and finite commutative chain rings. The structure of skewΘ-λ-constacyclic codes and their duals are provided. Among other results, we also consider the Euclidean and Hermitian dual codes of skewΘ-cyclic and skewΘ-negacyclic codes over finite chain rings in general and overFpm+uFpmin particular. Moreover, general decoding procedure for decoding skew BCH codes with designed distance and an algorithm for decoding skew BCH codes are discussed.

scholarly journals Constacyclic Codes over Finite Chain Rings of Characteristic p

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 303
Author(s):  
Sami Alabiad ◽  
Yousef Alkhamees
Keyword(s):  
Local Rings ◽  
Constacyclic Codes ◽  
Finite Chain ◽  
Dual Codes ◽  
Chain Ring ◽  
Characteristic P ◽  
Arbitrary Length ◽  
Cyclotomic Cosets ◽  
Finite Chain Rings ◽  
Polynomial Representations

Let R be a finite commutative chain ring of characteristic p with invariants p,r, and k. In this paper, we study λ-constacyclic codes of an arbitrary length N over R, where λ is a unit of R. We first reduce this to investigate constacyclic codes of length ps (N=n1ps,p∤n1) over a certain finite chain ring CR(uk,rb) of characteristic p, which is an extension of R. Then we use discrete Fourier transform (DFT) to construct an isomorphism γ between R[x]/<xN−λ> and a direct sum ⊕b∈IS(rb) of certain local rings, where I is the complete set of representatives of p-cyclotomic cosets modulo n1. By this isomorphism, all codes over R and their dual codes are obtained from the ideals of S(rb). In addition, we determine explicitly the inverse of γ so that the unique polynomial representations of λ-constacyclic codes may be calculated. Finally, for k=2 the exact number of such codes is provided.

scholarly journals Repeated-root constacyclic codes of prime power lengths over finite chain rings

2017 ◽  
Vol 43 ◽  
pp. 22-41 ◽  
Author(s):  
Hai Q. Dinh ◽  
Hien D.T. Nguyen ◽  
Songsak Sriboonchitta ◽  
Thang M. Vo
Keyword(s):  
Prime Power ◽  
Constacyclic Codes ◽  
Finite Chain ◽  
Finite Chain Rings

Structure of repeated-root constacyclic codes of length 8ℓmpn

2019 ◽  
Vol 12 (04) ◽  
pp. 1950050
Author(s):  
Saroj Rani
Keyword(s):  
Finite Field ◽  
Linear Codes ◽  
Important Class ◽  
Constacyclic Codes ◽  
Dual Codes ◽  
Negacyclic Codes ◽  
Positive Integers

Constacyclic codes form an important class of linear codes which is remarkable generalization of cyclic and negacyclic codes. In this paper, we assume that [Formula: see text] is the finite field of order [Formula: see text] where [Formula: see text] is a power of the prime [Formula: see text] and [Formula: see text] are distinct odd primes, and [Formula: see text] are positive integers. We determine generator polynomials of all constacyclic codes of length [Formula: see text] over the finite field [Formula: see text] We also determine their dual codes.

Negacyclic self-dual codes over finite chain rings

2011 ◽  
Vol 62 (2) ◽  
pp. 161-174 ◽  
Author(s):  
Xiaoshan Kai ◽  
Shixin Zhu
Keyword(s):  
Finite Chain ◽  
Dual Codes ◽  
Finite Chain Rings

On the depth spectrum of repeated-root constacyclic codes over finite chain rings

2020 ◽  
Vol 343 (2) ◽  
pp. 111647
Author(s):  
Jian Yuan ◽  
Shixin Zhu ◽  
Xiaoshan Kai
Keyword(s):  
Constacyclic Codes ◽  
Finite Chain ◽  
Finite Chain Rings

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].

Repeated-root constacyclic codes of arbitrary lengths over the Galois ring GR(p2,m)

2018 ◽  
Vol 10 (03) ◽  
pp. 1850036 ◽  
Author(s):  
Anuradha Sharma ◽  
Tania Sidana
Keyword(s):  
Positive Integer ◽  
Galois Ring ◽  
Constacyclic Codes ◽  
Dual Codes ◽  
Negacyclic Codes

Let [Formula: see text] be a prime, [Formula: see text] be a positive integer, and let GR[Formula: see text] be the Galois ring of characteristic [Formula: see text] and cardinality [Formula: see text]. In this paper, all repeated-root constacyclic codes of arbitrary lengths over GR[Formula: see text] their sizes and their dual codes are determined. As an application, some isodual constacyclic codes over GR[Formula: see text] are also listed. To illustrate the results, all cyclic and negacyclic codes of length 10 over [Formula: see text] are obtained.

scholarly journals Some New Quantum BCH Codes over Finite Fields

Entropy ◽  
2021 ◽  
Vol 23 (6) ◽  
pp. 712
Author(s):  
Lijuan Xing ◽  
Zhuo Li
Keyword(s):  
Finite Fields ◽  
Error Correcting Codes ◽  
Bch Codes ◽  
Quantum Codes ◽  
Dual Codes ◽  
Cyclotomic Cosets ◽  
New Family ◽  
Consecutive Integers ◽  
Css Construction ◽  
Quantum Error

Quantum error correcting codes (QECCs) play an important role in preventing quantum information decoherence. Good quantum stabilizer codes were constructed by classical error correcting codes. In this paper, Bose–Chaudhuri–Hocquenghem (BCH) codes over finite fields are used to construct quantum codes. First, we try to find such classical BCH codes, which contain their dual codes, by studying the suitable cyclotomic cosets. Then, we construct nonbinary quantum BCH codes with given parameter sets. Finally, a new family of quantum BCH codes can be realized by Steane’s enlargement of nonbinary Calderbank-Shor-Steane (CSS) construction and Hermitian construction. We have proven that the cyclotomic cosets are good tools to study quantum BCH codes. The defining sets contain the highest numbers of consecutive integers. Compared with the results in the references, the new quantum BCH codes have better code parameters without restrictions and better lower bounds on minimum distances. What is more, the new quantum codes can be constructed over any finite fields, which enlarges the range of quantum BCH codes.

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