The Gray image of constacyclic codes over the finite chain ring $$F_{p^m}[u]/\langle u^k\rangle $$ F p m [ u ] / ⟨ u k ⟩

2017 ◽  
Vol 57 (1-2) ◽  
pp. 303-320 ◽  
Author(s):  
Yuan Cao ◽  
Yonglin Cao
Keyword(s):  
Constacyclic Codes ◽  
Finite Chain ◽  
Chain Ring ◽  
Finite Chain Ring ◽  
Gray Image

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 On the Hamming distance of linear codes over a finite chain ring

2000 ◽  
Vol 46 (3) ◽  
pp. 1060-1067 ◽  
Author(s):  
G.H. Norton ◽  
A. Salagean
Keyword(s):  
Linear Codes ◽  
Hamming Distance ◽  
Finite Chain ◽  
Chain Ring ◽  
Finite Chain Ring

On repeated-root multivariable codes over a finite chain ring

2007 ◽  
Vol 45 (2) ◽  
pp. 219-227 ◽  
Author(s):  
E. Martínez-Moro ◽  
I. F. Rúa
Keyword(s):  
Finite Chain ◽  
Chain Ring ◽  
Finite Chain Ring

scholarly journals Communication Over Finite-Chain-Ring Matrix Channels

2014 ◽  
Vol 60 (10) ◽  
pp. 5899-5917 ◽  
Author(s):  
Chen Feng ◽  
Roberto W. Nobrega ◽  
Frank R. Kschischang ◽  
Danilo Silva
Keyword(s):  
Finite Chain ◽  
Chain Ring ◽  
Finite Chain Ring

Classification of All Cyclic Codes over the Finite Chain Ring Z4+wZ4 of Length 4s

2013 ◽  
Vol 13 (7) ◽  
pp. 1037-1044
Author(s):  
Xiaofang Xu ◽  
Shujie Yun
Keyword(s):  
Cyclic Codes ◽  
Finite Chain ◽  
Chain Ring ◽  
Finite Chain Ring

CAYLEY GRAPHS OVER A FINITE CHAIN RING AND GCD-GRAPHS

2016 ◽  
Vol 93 (3) ◽  
pp. 353-363 ◽  
Author(s):  
BORWORN SUNTORNPOCH ◽  
YOTSANAN MEEMARK
Keyword(s):  
Graph Theory ◽  
Prime Power ◽  
Quotient Ring ◽  
Spectral Graph Theory ◽  
Prime Power Order ◽  
Finite Chain ◽  
Circulant Graphs ◽  
Chain Ring ◽  
Finite Chain Ring ◽  
Spectral Graph

We extend spectral graph theory from the integral circulant graphs with prime power order to a Cayley graph over a finite chain ring and determine the spectrum and energy of such graphs. Moreover, we apply the results to obtain the energy of some gcd-graphs on a quotient ring of a unique factorisation domain.

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.

Quantum codes over a class of finite chain ring

2016 ◽  
pp. 39-49
Author(s):  
Mustafa Sari ◽  
Irfan Siap
Keyword(s):  
Finite Field ◽  
Cyclic Codes ◽  
Gray Map ◽  
Quantum Codes ◽  
Finite Chain ◽  
Chain Ring ◽  
Finite Chain Ring ◽  
Quantum Error

In this study, we introduce a new Gray map which preserves the orthogonality from the chain ring F_2 [u] / (u^s ) to F^s_2 where F_2 is the finite field with two elements. We also give a condition of the existence for cyclic codes of odd length containing its dual over the ring F_2 [u] / (u^s ) . By taking advantage of this Gray map and the structure of the ring, we obtain two classes of binary quantum error correcting (QEC) codes and we finally illustrate our results by presenting some examples with good parameters.

New quantum codes from cyclic codes over a finite chain ring of length 3

2020 ◽  
Author(s):  
Brahim Boudine ◽  
Jamal Laaouine ◽  
Mohammed Elhassani Charkani
Keyword(s):  
Cyclic Codes ◽  
Quantum Codes ◽  
Finite Chain ◽  
Chain Ring ◽  
Finite Chain Ring
Sign in / Sign up

Export Citation Format

Share Document