Cyclic codes and λ1+λ2u+λ3v+λ4uv-constacyclic codes over Fp+uFp+vFp+uvFp

doi:10.1016/j.amc.2017.02.017

Applied Mathematics and Computation

Volume 306, 1 August 2017, Pages 86-91

https://doi.org/10.1016/j.amc.2017.02.017 Get rights and content

Abstract

Let $R = F_{p} + u F_{p} + v F_{p} + u v F_{p},$ where p is a prime and $u^{2} = u,$ $v^{2} = v,$ $u v = v u$ . It is a semi-local ring, not a chain ring. In this paper, some properties of linear codes over the ring R are studied. The generator of cyclic codes and $λ_{1} + λ_{2} u + λ_{3} v + λ_{4} u v$ -constacyclic codes over R are given. A necessary and sufficient condition for self-dual cyclic codes over R is given.

Introduction

Constacyclic codes are an important class of linear codes and have great interest in coding theory. And, constacyclic codes also have practical applications as they can be encoded with shift registers. Cengellenmis in [1] studied the cyclic codes over $F_{3} + v F_{3}$ . Gao in [2] studied some properties of linear codes over the ring $F_{p} + u F_{p} + u^{2} F_{p}$ . Dinh et al. in [3] studied the structures of all negacyclic codes of length 2p^s over $F_{p^{m}} + u F_{p^{m}}$ . Wang and Zhu in [4] gave a way to construct quantum MDS codes based on classical constacyclic codes over $F_{q^{2}}$ . Hu et al. in [5] gave a construction of two new families of good nonbinary quantum codes. Kai et al. in [6] constructed two new classes of quantum MDS codes based on classical negacyclic codes. Alahmadia et al. in [7] constructed cyclic isodual codes over $F_{q}$ . Ashraf and Mohammad in [8] gave a construction of quantum codes from cyclic codes over ring $F_{3} + v F_{3}$ with $v^{2} = 1$ . Dertli et al. in [9] gave a construction of quantum codes from cyclic codes over $F_{2} + v F_{2} + v^{2} F_{2}$ with $v^{2} = v$ . Chapman et al. in [10] studied 2-modular lattices from ternary codes. Bandi and Bhaintwal in [11] studied cyclic codes over the ring $Z_{4} + u Z_{4}$ with $u^{2} = 0$ obtained a minimal spanning set and ranks for cyclic codes over the ring. Guenda and Gulliver in [12] gave a construction of quantum error correcting from linear codes over finite commutative Frobenius rings. Kai and Zhu in [13] gave a construction for quantum codes from linear and cyclic codes over $F_{4} + u F_{4}$ with $u^{2} = 0$ . In [14], Cengellenmis et al. gave the structure of cyclic codes and their Gray map. Constacyclic codes over a generalization of this ring, namely $F_{2} + u F_{2} + v F_{2} + u v F_{2}$ were introduced by Karadeniz and Yildiz in [15]. In [16], the cyclic codes over the ring $Z_{p} [u, v] / 〈 u^{2}, v^{2}, u v - v u 〉$ were studied. In [17], quantum codes from cyclic codes over $F_{2} + u F_{2} + v F_{2} + u v F_{2}$ for arbitrary length n were constructed. In [18], Ma et al. determined the sharp upper bound of the Wiener polarity index among all bicyclic networks. In [19], some results for chemical trees were obtained.

The remainder of this paper is organized as follows. In Section 2, we discuss the preliminaries that we need. In Section 3, we define a Gray map from R to $F_{p}^{4}$ . In Section 4, we give the structure of cyclic codes of arbitrary length over R and we give a necessary and sufficient condition for self-dual cyclic codes. In Section 5, we give the structure of $λ_{1} + λ_{2} u + λ_{3} v + λ_{4} u v$ -constacyclic codes of arbitrary length over R.

Section snippets

Preliminaries

An ideal I of a finite commutative ring R is called principal if it is generated by one element. R is a principal ideal ring if its ideas are principal. R is called a local ring if R has a unique maximal ideal. R is called a chain ring if its ideals are linearly ordered by inclusion.

We recall that a linear code C of length n over R is just a R-submodule of Rⁿ. Every codeword c is a n-tuple of the form $c = (c_{0}, c_{1}, \dots, c_{n - 1}) \in R^{n}$ and can be represented by a polynomial in R[x] as follows: $c = (c_{0}, c_{1}, \dots, c_{n - 1})$

Gray map

Again as in [15], we define the Gray map $ϕ : R \to F_{p}^{4}$ as follows $ϕ (a + u b + v c + u v d) = (a, a + b, a + c, a + b + c + d), \forall a, b, c, d \in F_{p} .$

Every element r of R can be represented as $r = a + b u + c v + d u v,$ where $a, b, c, d \in F_{p}$ . The Gray map naturally extends to Rⁿ as distance preserving isometry

$ϕ : (R^{n}, Lee weight) \to (F_{p}^{4 n}, Hamming weight)$ as follows:

$ϕ (r_{1}, r_{2}, \dots, r_{n}) = (a_{1}, \dots, a_{n}, a_{1} + b_{1}, \dots, a_{n} + b_{n}, a_{1} + c_{1}, \dots, a_{n} + c_{n}, a_{1} + b_{1} + c_{1} + d_{1}, \dots, a_{n} + b_{n} + c_{n} + d_{n}),$ $\forall r_{i} = a_{i} + b_{i} u + c_{i} v + d_{i} u v \in R^{n},$ where $a_{i}, b_{i}, c_{i}, d_{i} \in F_{p}$ .

By the definition of the Gray map we can get the following theorem easily:

Theorem 1

Cyclic codes over R

As in [17] we denote that $A_{1} \otimes A_{2} \otimes A_{3} \otimes A_{4} = {(a_{1}, a_{2}, a_{3}, a_{4}), a_{i} \in A_{i}, i = 1, 2, 3, 4},$ $A_{1} \oplus A_{2} \oplus A_{3} \oplus A_{4} = {a_{1} + a_{2} + a_{3} + a_{4}, a_{i} \in A_{i}, i = 1, 2, 3, 4}$ .

For all $a, b, c, d \in F_{p},$ $a + u b + v c + u v d = a (1 - u - v + u v) + (a + b) (u - u v) + (a + c) (v - u v) + (a + b + c + d) u v$ . Let C be a linear code of length n over R, let

$C_{1} = {a \in F_{p}^{n} | a + u b + v c + u v d \in C, \exists b, c, d \in F_{p}^{n}},$

$C_{2} = {a + b \in F_{p}^{n} | a + u b + v c + u v d \in C, \exists c, d \in F_{p}^{n}},$

$C_{3} = {a + c \in F_{p}^{n} | a + u b + v c + u v d \in C, \exists b, d \in F_{p}^{n}},$

$C_{4} = {a + b + c + d \in F_{p}^{n} | a + u b + v c + u v d \in C},$

Obviously, C₁, C₂, C₃ and C₄ are linear codes over $F_{p}$ .

Moreover, the linear code C can be written as $C = (1 - u - v + u v) C_{1} \oplus (u - u v) C_{2}$

$λ_{1} + λ_{2} u + λ_{3} v + λ_{4} u v$ -constacyclic codes over R

Theorem 7

If $C = (1 - u - v + u v) C_{1} \oplus (u - u v) C_{2} \oplus (v - u v) C_{3} \oplus (u v) C_{4}$ is a linear code over R, then C is a $λ_{1} + λ_{2} u + λ_{3} v + λ_{4} u v$ -constacyclic code over R if and only if C₁, C₂, C₃, C₄ are λ₁-constacyclic code, $λ_{1} + λ_{2}$ -constacyclic code, $λ_{1} + λ_{3}$ -constacyclic code, $λ_{1} + λ_{2} + λ_{3} + λ_{4}$ -constacyclic code over $F_{p},$ respectively, where $λ_{1} + λ_{2} u + λ_{3} v + λ_{4} u v$ is a unit over R.

Proof

For any $r = (r_{0}, r_{1}, \dots, r_{n - 1}) \in C,$ we can write its components as $r_{i} = (1 - u - v + u v) a_{i} + (u - u v) b_{i} + (v - u v) c_{i} + u v d_{i},$ where $a_{i}, b_{i}, c_{i}, d_{i} \in F_{p},$ $0 \leq i \leq n - 1$ . Let $a = (a_{0}, a_{1}, \dots, a_{n - 1}), b = (b_{0}, b_{1}, \dots, b_{n - 1}), c = (c_{0}, c_{1}, \dots, c_{n - 1}), d = ($

Conclusion

In this paper, we studied the cyclic codes and $λ_{1} + λ_{2} u + λ_{3} v + λ_{4} u v$ -constacyclic codes over the finite non-chain ring $F_{p} + u F_{p} + v F_{p} + u v F_{p},$ where $u^{2} = u, v^{2} = v$ and $u v = v u$ . We gave a sufficient and necessary condition for self-dual cyclic codes.

Acknowledgments

This work was supported by the Key Project of Science and Technology of Zhengzhou (no. 20141375), the Basic and Advanced Technology Research project of Henan Province (no. 162300410083) and the Science and Technology Developing Project of Henan Province (no. 172102210243).

References (19)

H.Q. Dinh et al.
Negacyclic codes of length 2p^s over $f_{p^{m}} + u f_{p^{m}}$
Finite Fields Appl.
(2015)
Y. Shi
Note on two generalizations of the Randic index
Appl. Math. Comput.
(2015)
Y. Cengellenmis
On the cyclic codes over $f_{3} + v f_{3}$
Int. J. Algebra
(2010)
J. Gao
Some results on linear codes over $f_{p} + u f_{p} + u^{2} f_{p}$
J. Appl. Math. Comput.
(2015)
L. Wang et al.
New quantum MDS codes derived from constacyclic codes
Quant. Inf. Process.
(2015)
X. Hu et al.
Constructions of new nonbinary quantum codes
Int. J. Theor. Phys.
(2015)
X. Kai et al.
Constacyclic codes and some new quantum MDS codes
IEEE Trans. Inf. Theory
(2014)
A. Alahmadia et al.
Isodual cyclic codes over finite fields of odd characteristic
Discrete Math.
(2016)
M. Ashraf et al.
Quantum codes from cyclic codes over $f_{3} + v f_{3}$
Int. J. Quant. Inf.
(2014)

There are more references available in the full text version of this article.

Cited by (13)

Cyclic codes over a non-chain ring R<inf>e,q</inf> and their application to LCD codes
2021, Discrete Mathematics
Let $F_{q}$ be a finite field of order q, a prime power integer, such that $q = e t + 1$ where $t \geq 1, e \geq 2$ are integers. In this paper, we study cyclic codes of length n over a non-chain ring $R_{e, q} = F_{q} [u] / 〈 u^{e} - 1 〉$ . We define a Gray map φ and obtain many maximum-distance-separable (MDS) and optimal $F_{q}$ -linear codes from the Gray images of cyclic codes. Under certain conditions we determine linear complementary dual (LCD) codes of length n when $\gcd (n, q) \neq 1$ and $\gcd (n, q) = 1$ , respectively. It is proved that a cyclic code $C$ of length n is an LCD code if and only if its Gray image $φ (C)$ is an LCD code of length en over $F_{q}$ . Among others, we present the conditions for existence of free and non-free LCD codes. Moreover, we obtain many optimal LCD codes as the Gray images of non-free LCD codes over $R_{e, q}$ .
Quantum codes from constacyclic codes over S<inf>k</inf>
2023, EPJ Quantum Technology
NEW QUANTUM CODES FROM SKEW CONSTACYCLIC CODES
2023, Advances in Mathematics of Communications
Constacyclic and Skew Constacyclic Codes Over a Finite Commutative Non-chain Ring
2023, Springer Proceedings in Mathematics and Statistics
(θ, δ<inf>θ</inf>) -Cyclic codes over F<inf>q</inf>[u, v] / ⟨ u<sup>2</sup>- u, v<sup>2</sup>- v, uv- vu⟩
2022, Designs, Codes, and Cryptography
A note on two-dimensional cyclic and constacyclic codes
2022, Journal of Algebra Combinatorics Discrete Structures and Applications

View all citing articles on Scopus

View full text

Cyclic codes and λ1+λ2u+λ3v+λ4uv-constacyclic codes over Fp+uFp+vFp+uvFp

Abstract

Introduction

Section snippets

Preliminaries

Gray map

Cyclic codes over R

λ1+λ2u+λ3v+λ4uv-constacyclic codes over R

Conclusion

Acknowledgments

Finite Fields Appl.

Appl. Math. Comput.

On the cyclic codes over f3+vf3

Int. J. Algebra

Some results on linear codes over fp+ufp+u2fp

J. Appl. Math. Comput.

New quantum MDS codes derived from constacyclic codes

Quant. Inf. Process.

Constructions of new nonbinary quantum codes

Int. J. Theor. Phys.

Constacyclic codes and some new quantum MDS codes

IEEE Trans. Inf. Theory

Isodual cyclic codes over finite fields of odd characteristic

Discrete Math.

Quantum codes from cyclic codes over f3+vf3

Int. J. Quant. Inf.

Cyclic codes and $λ_{1} + λ_{2} u + λ_{3} v + λ_{4} u v$ -constacyclic codes over $F_{p} + u F_{p} + v F_{p} + u v F_{p}$

$λ_{1} + λ_{2} u + λ_{3} v + λ_{4} u v$ -constacyclic codes over R

On the cyclic codes over $f_{3} + v f_{3}$

Some results on linear codes over $f_{p} + u f_{p} + u^{2} f_{p}$

Quantum codes from cyclic codes over $f_{3} + v f_{3}$