Skew Cyclic and Skew Constacyclic Codes over a Mixed Alphabet

Karthick Gowdhaman

¹,

Cruz Mohan

^2,*

Chinnapillai Durairajan

³,

Selda Çalkavur

⁴ and

Patrick Solé

⁵

Department of Mathematics, Presidency University, Bengaluru 560064, Karnataka, India

Department of Mathematics, Bishop Heber College, Bharathidasan University, Tiruchirapalli 620017, Tamil Nadu, India

Department of Mathematics, Bharathidasan University, Tiruchirapalli 620024, Tamil Nadu, India

⁴

Department of Mathematics, Faculty of Arts and Science, Kocaeli University, Kocaeli 41001, Turkey

⁵

I2M, Aix Marseille University, CNRS, 13009 Marseille, France

Author to whom correspondence should be addressed.

Axioms2024,13(6), 360;https://doi.org/10.3390/axioms13060360

Submission received: 3 March 2024 /Revised: 20 April 2024 /Accepted: 22 May 2024 /Published: 28 May 2024

Downloadkeyboard_arrow_downVersions Notes

Abstract

In this note, we study skew cyclic and skew constacyclic codes over the mixed alphabet

R = F_{q} R_{1} R_{2}

, where

q = p^{m},

p is an odd prime withm odd and

R_{1} = F_{q} + u F_{q}

with

u^{2} = u

, and

R_{2} = F_{q} + u F_{q} + v F_{q}

with

u^{2} = u, v^{2} = v, u v = v u = 0 .

Such codes consist of the juxtaposition of three codes of the same size over

F_{q},

R_{1},

and

R_{2}

, respectively. We investigate the generator polynomial for skew cyclic codes over

R

. Furthermore, we discuss the structural properties of the skew cyclic and skew constacyclic codes over

R .

We also study theirq-ary images under suitable Gray maps.

Keywords:

linear codes;skew-cyclic codes;Gray map;skew constacyclic codes

MSC:

94B05; 94B15; 94B35; 94B60

1. Introduction

The most widely used family of linear codes consists of cyclic codes. Inspired by codes for the Lee metric [1], Berlekamp adapted them to constacyclic codes. Since then, as the following paragraphs demonstrate, they have happened in a number of circumstances.

Skew cyclic codes were first introduced as ideals in the skew polynomial ring

F [x; θ]

in 2007 by Boucher et al. [2], where

θ

represents an automorphism of the finite field

F .

The tables of the most well-known codes were enhanced by the numerous numerical samples that this technique created. The fact that the factorization of the polynomial

x^{n} - 1

is not unique gives skew polynomial rings an advantage over commutative polynomial rings. For a given length, these numerous factorizations produce a large number of additional codes. Boucher et al. further extended this technique to skew constacyclic codes in [3]. Siap et al. [4] investigated skew cyclic codes of any length in 2011 and produced maps using both classical and quasi-cyclic codes.

In 2012, Jitman et al. [5] studied skew constacyclic codes over finite chain rings and described the algebraic structure of Euclidean and Hermitian dual codes. Abualrub et al. [6] studied

θ

-cyclic codes over the semilocal ring

F_{2} + u F_{2}, v^{2} = v

with respect to Euclidean and Hermitian inner products.

These codes over semilocal rings were further studied in many contexts. The rings

F_{3} + v F_{3}

in [7],

F_{q} + v F_{q}, v^{2} = v

in [8], and

F_{q} + u F_{q} + v F_{q}, u^{2} = u, v^{2} = v, u v = v u = 0

in [9] were utilized as alphabets for skew cyclic codes, for example. Dertli and Cengellenmis [10] and Yao et al. [11] also examined these codes over

F_{q} + u F_{q} + v F_{q} + u v F_{q}, u^{2} = u, v^{2} = v,

u v = v u

. Skewed

(- 1 + 2 v)

-constacyclic codes were developed in 2017 by Gao et al. [12] after deriving the structure of skew constacyclic codes over the semilocal ring

F_{q} + v F_{q}, v^{2} = v

. In [13] and [14], respectively, Islam and Prakash established the algebraic structure of skew constacyclic codes over

F_{q} + u F_{q} + v F_{q} + u v F_{q}, u^{2} = u, v^{2} = v, u v = v u

Using two non-trivial automorphisms, Bhardwaj and Raka [15] investigated the skew constacyclic codes over the ring

F_{q} [u, v] 〈 f (u), g (v), u v - v u 〉

in 2019. Alternatively,

Z_{2} Z_{4}

-linear codes, or codes over the mixed alphabet

Z_{2} Z_{4},

where a subset of coordinates is binary and the complement is quaternary, were introduced by Borges et al. [16]. They have calculated their generator matrices and described their dual codes. Fernandez-Cordoba et al. [17] obtained the rank and kernel of

Z_{2} Z_{4}

-linear codes in a follow-up experiment. Steganography is one field in which these codes have found industrial use [18].

In [19], additive codes over the mixed alphabet

Z_{2} Z_{2^{s}}

were examined. Next, Refs. [20,21,22,23] examined the mixed alphabet

Z_{p} Z_{p^{s}}

and, more broadly,

Z_{p^{r}} Z_{p^{s}}

. Conversely, Abualrub et al. [24] defined

Z_{2} Z_{4}

in 2014, in line with the advancement of cyclic codes on mixed alphabets. The code for -additive cyclics is

Z_{4} [x]

.-submodule of

Z_{2} [x] / 〈 x^{r} - 1 〉 \times Z_{4} [x] / 〈 x^{s} - 1 〉

, from which the smallest spanning set and unique set of generators for these codes, wheres is an odd integer, were obtained.

Furthermore, generator polynomials and duals for

Z_{2} Z_{4}

-additive cyclic codes were discovered by Borges et al. [25]. In [26], Aydogdu et al. [27] introduced the novel mixed alphabets

Z_{2} Z_{2} [u]

-additive codes, where

u^{2} = 0

. They also studied constacyclic codes over mixed alphabets by defining them as

Z_{2} [u] [x] . \times Z_{2} [u] [x] / 〈 x^{β} - (1 + u) 〉

, -submodules of

Z_{2} [x] / 〈 x^{α} - 1 〉

As the Gray images of

Z_{2} Z_{2} [u]

-cyclic codes, they were able to derive several optimum binary linear codes. In the meanwhile,

Z_{2} Z_{2} [u]

-additive cyclic and constacyclic codes with the unit

1 + u

, respectively, were explored algebraically by [28]. Consequently, the predicted generalization in the continuation of these research should be

Z_{2^{r}} Z_{2^{s}} [u]

.-additive cyclic codes,

u^{2} = 0

, and constacyclic codes.

In this article, we examine a mixed alphabet

R = F_{q} R_{1} R_{2}

, where

R_{1} = F_{q} + u F_{q}

with

u^{2} = u

and

R_{2} = F_{q} + u F_{q} + v F_{q}

with

u^{2} = u, v^{2} = v, u v = v u = 0 .

Moreover, we examine the cyclic codes

θ_{t}

and

(θ_{t}, α)

over

R .

The algebraic structure of these codes is fully determined. We examine theirq-ary representations under Gray maps and provide a few brief numerical instances.

The contents are arranged as follows. The next section gathers some background information. Gray maps are examined inSection 3. Skewed cyclic codes are covered inSection 4, and skew constacyclic codes are covered inSection 5. The essay is concluded atSection 6.

2. Preliminaries

Letp be an odd prime, and let

q = p^{m}

withm being odd. Denote by

F_{q}

the finite field of size

q .

The set

F_{q}^{n}

of all orderedn-tuples over

F_{q}

is equipped with the structure of an

F_{q}

vector space by the usual addition and scalar multiplication of vectors.

A code of lengthn over

F_{q}

is just any non-empty subset

C of F_{q}^{n} .

It is said to be linear if

C

is an

F_{q}

subspace of

F_{q}^{n} .

From now on, we write

R_{1} = F_{q} + u F_{q},

with

u^{2} = u

and

R_{2} = F_{q} + u F_{q} + v F_{q},

with

u^{2} = u, v^{2} = v, u v = v u = 0 .

Note that

R_{1}

and

R_{2}

are finite non-chain rings. Let

a + u b + v c

be an element of

R_{2} .

Then, we define two maps

η

and

δ

as follows:

\begin{matrix} η : R_{2} \to F_{q}, & δ : R_{2} \to R_{1}, \\ η (a + u b + v c) = a, & δ (a + u b + v c) = a + u b, \end{matrix}

It is clear that

η and δ

are ring homomorphisms. We consider the ring

R

R = F_{q} R_{1} R_{2} = {(x, y, z) ∣ x \in F_{q}, y \in R_{1} and z \in R_{2}}

We define a

R_{2}

-multiplication in this ring as follows:

\begin{matrix} ★ : R_{2} \times R & \to & R \\ r ★ (x, y, z) & = & (η (r) x, δ (r) y, r z) \end{matrix}

This is a well defined multiplication and it can be extended componentwise to

R_{⚪} = F_{q}^{n_{1}} \times R_{1}^{n_{2}} \times R_{2}^{n_{3}}

by:

\begin{matrix} ★ : R_{2} \times R_{⚪} & \to & R_{⚪} \\ r ★ (x_{1}, \dots, x_{n_{1}}, y_{1}, \dots, y_{n_{2}}, z_{1}, \dots, z_{n_{3}}) & = & (η (r) x_{1}, \dots, η (r) x_{n_{1}}, δ (r) y_{1}, \dots, δ (r) y_{n_{2}}, r z_{1}, \dots, r z_{n_{3}}) \end{matrix}

where

(x_{1}, \dots, x_{n_{1}}, y_{1}, \dots, y_{n_{2}}, z_{1}, \dots, z_{n_{3}}) \in R_{γ} .

Equipped with this multiplication,

R_{⚪}

becomes an

R_{2} module .

A non-empty subset

C of R_{⚪}

is said to be a

R

-linear code of length

(n_{1}, n_{2}, n_{3})

C is an R_{2} - submodule of R_{⚪} .

Now we define the inner product by the formula:

\begin{matrix} 〈 c, c^{'} 〉 = \sum_{1}^{n_{1}} x_{i} x_{i}^{'} + \sum_{1}^{n_{2}} y_{j} y_{j}^{'} + \sum_{1}^{n_{3}} z_{k} z_{k}^{'}, \end{matrix}

where

c = (x_{1}, \dots, x_{n_{1}}, y_{1}, \dots, y_{n_{2}}, z_{1}, \dots, z_{n_{3}}), c^{'} = (x_{1}^{'}, \dots, x_{n_{1}}^{'}, y_{1}^{'}, \dots, y_{n_{2}}^{'}, z_{1}^{'}, \dots, z_{n_{3}}^{'})

are in

R_{γ} .

Let

C

be an

R

-linear code of length

(n_{1}, n_{2}, n_{3}) .

Then, the dual code of

C

is defined as:

\begin{matrix} C^{⊥} = {c^{'} \in R_{γ} ∣ 〈 c, c^{'} 〉 = 0 \forall c \in C} \end{matrix}

3. Decomposition and Properties of Gray Maps

Recall that,

R_{1} = F_{q} + u F_{q}, with u^{2} = u .

Consider the idempotent orthogonal elements

e_{1} = u

and

e_{2} = 1 - u .

Then, we have the decomposition:

R_{1} = e_{1} R_{1} \oplus e_{2} R_{1} ≅ e_{1} F_{q} \oplus e_{2} F_{q},

where

e_{1} e_{2} = 0, e_{1}^{2} = e_{1}, e_{1} + e_{2} = 1 .

Hence,

R_{1} = {a e_{1} + b e_{2} ∣ a, b in F_{q}} .

We now define the Gray map:

\begin{matrix} φ_{1} : R_{1} & \to & F_{q}^{2} \\ φ_{1} (a e_{1} + b e_{2}) & = & (a, b) \end{matrix}

It can be extended to the lengthn by:

\begin{matrix} φ_{1} : R_{1}^{n} & \to & F_{q}^{2 n} \\ φ_{1} ((a_{1}, \dots, a_{n}) e_{1} + (b_{1}, \dots, b_{n}) e_{2}) & = & (a_{1}, \dots, a_{n}, b_{1}, \dots, b_{n}) \end{matrix}

Note that it is a linear map. We define the Gray weight of a codeword in

R_{1}

as:

w t_{G} (a e_{1} + b e_{2}) = w t_{H} (a, b)

where

w t_{H}

denotes the Hamming weight. If

x, y lie in R_{1}^{n},

then their mutual distance is given by:

d_{G} (x, y) = \sum_{1}^{n} w t_{G} (x_{i} - y_{i}) = \sum_{1}^{2 n} w t_{H} (φ_{1} (x) - φ_{1} (y)) = d_{H} (φ_{1} (x), φ_{1} (y)) .

Hence,

φ_{1}

is a weight preserving map. A non-empty subset

C of R_{i}^{n}

is said to be a linear code of lengthn if

C

R_{i} - submodule of R_{i}^{n} .

For

i \in {1, 2}, A_{i} \subseteq R_{1}

A_{1} \oplus A_{2} = {a_{1} + a_{2} ∣ a_{i} \in A_{i}} and A_{1} \otimes A_{2} = {(a_{1}, a_{2}) ∣ a_{i} \in A_{i}} .

Let

C_{e}

be a linear code of lengthn over

R_{1} .

Then, we define:

C_{e_{1}} = {y_{1} \in F_{q}^{n} ∣ e_{1} y_{1} + e_{2} y_{2} \in C_{e}, for some y_{2} \in F_{q}^{n}}

C_{e_{2}} = {y_{2} \in F_{q}^{n} ∣ e_{1} y_{1} + e_{2} y_{2} \in C_{e}, for some y_{1} \in F_{q}^{n}}

Therefore, any linear code

C_{e}

over

R_{1}

can be represented as

C_{e} = e_{1} C_{e_{1}} \oplus e_{2} C_{e_{2}}

and

φ_{1} (C_{e}) = C_{e_{1}} \otimes C_{e_{2}} .

Hence,

C_{e_{1}}

and

C_{e_{2}}

are

F_{q}

-linear codes. Also note that

φ_{1} (C_{e}^{⊥}) = φ_{1} {(C_{e})}^{⊥} .

Recall that

R_{2} = F_{q} + u F_{q} + v F_{q},

with

u^{2} = u, v^{2} = v, u v = v u = 0 .

Let

o_{1} = (1 - u - v), o_{2} = u, o_{3} = v

be idempotent orthogonal elements in

R_{2},

then:

R_{2} = o_{1} R_{2} \oplus o_{2} R_{2} \oplus o_{3} R_{2} ≅ o_{1} F_{q} \oplus o_{2} F_{q} \oplus o_{2} F_{q},

where

o_{i} o_{j} = 0 (i \neq j), o_{i}^{2} = o_{i}, o_{1} + o_{2} + o_{3} = 1 .

Hence, any element in

R_{2}

can be written as

a o_{1} + b o_{2} + o_{3} c_{3} .

We now define a weight preserving linear Gray map

φ_{2},

\begin{matrix} φ_{2} : R_{2} & \to & F_{q}^{3} \\ φ_{2} (a o_{1} + b o_{2} + c o_{3}) & = & (a, b, c) \end{matrix}

It can be extended to lengthn by the formula:

\begin{matrix} φ_{2} ((a_{1}, \dots, a_{n}) o_{1} + (b_{1}, \dots, b_{n}) o_{2} + (c_{1}, \dots, c_{n}) o_{2}) = (a_{1}, \dots, a_{n}, b_{1}, \dots, b_{n}, c_{1}, \dots, c_{n}) . \end{matrix}

We define the Gray weight of a codeword in

R_{2}

as:

w t_{G} (a o_{1} + b o_{2} + c o_{3}) = w t_{H} (a, b, c)

where

w t_{H}

denotes the Hamming weight. If

x, y are in R_{2}^{n},

then their Gray distance is given by:

d_{G} (x, y) = \sum_{1}^{n} w t_{G} (x_{i} - y_{i}) = \sum_{1}^{3 n} w t_{H} (φ_{2} (x) - φ_{2} (y)) = d_{H} (φ_{2} (x), φ_{2} (y)) .

For

i \in {1, 2, 3}, A_{i} \subseteq R_{2}

A_{1} \oplus A_{2} \oplus A_{3} = {a_{1} + a_{2} + a_{3} ∣ a_{i} \in A_{i}} and A_{1} \otimes A_{2} \otimes A_{3} = {(a_{1}, a_{2}, a_{3}) ∣ a_{i} \in A_{i}} .

Let

C_{o}

be a linear code of lengthn over

R_{2} .

We define the three codes:

\begin{matrix} C_{o_{1}} & = & {z_{1} \in F_{q}^{n} ∣ o_{1} z_{1} + o_{2} z_{2} + o_{3} z_{3} \in C_{o}, for some z_{2}, z_{3} \in F_{q}^{n}}, \\ C_{o_{2}} & = & {z_{2} \in F_{q}^{n} ∣ o_{1} z_{1} + o_{2} z_{2} + o_{3} z_{3} \in C_{o}, for some z_{1}, z_{3} \in F_{q}^{n}}, \\ C_{o_{3}} & = & {z_{3} \in F_{q}^{n} ∣ o_{1} z_{1} + o_{2} z_{2} + o_{3} z_{3} \in C_{o}, for some z_{1}, z_{2} \in F_{q}^{n}} . \end{matrix}

Then, any linear code

C_{o}

over

R_{2}

can be represented as

C_{o} = o_{1} C_{o_{1}} \oplus o_{2} C_{o_{2}} \oplus o_{3} C_{o_{3}}

and

φ_{2} (C_{o}) = C_{o_{1}} \otimes C_{o_{2}} \otimes C_{o_{3}}, where C_{o_{1}}, C_{o_{2}}

, and

C_{o_{3}}

are

F_{q}

-linear codes. Also note that

φ_{2} (C_{o}^{⊥}) = φ_{2} {(C_{o})}^{⊥} .

Henceforth, we define the Gray map

φ on R

using the maps defined previously:

\begin{matrix} φ : R & \to & F_{q}^{6} \\ φ (x, y, z) & = & (x, φ_{1} (y), φ_{2} (z)) \end{matrix}

now we can extend this map to

R_{γ}

\begin{matrix} φ (x_{1}, \dots, x_{n_{1}}, y_{1}, \dots, y_{n_{2}}, z_{1}, \dots, z_{n_{3}}) = (x_{1}, \dots, x_{n_{1}}, φ_{1} (y_{1}), \dots, φ_{1} (y_{n_{2}}), φ_{2} (z_{1}), \dots, φ_{2} (z_{n_{3}})) \end{matrix}

then, the Gray weight of an element in

R_{γ}

can be denoted by

w t_{G} (α) = w t_{H} (φ (α))

. Any linear code

C

R_{γ}

can be represented by

C = C_{1} \otimes C_{e} \otimes C_{o},

where

C_{1}, C_{e}, and C_{o}

are linear code over

F_{q}, R_{1}, and R_{2} .

Let

G_{F_{q}}

be the generator matrix for linear code over

F_{q} .

The generator matrix

G_{R_{1}}

for a linear code over

R_{1}

is denoted by:

G_{R_{1}} = [\begin{matrix} e_{1} G_{e_{1}} \\ e_{2} G_{e_{2}} \end{matrix}]

where

G_{e_{i}}

is the generator matrix for the linear code

{C_{e}}_{i}, for i = {1, 2} .

The generator matrix

G_{R_{2}}

for the linear code over

R_{2}

is:

G_{R_{2}} = [\begin{matrix} o_{1} G_{o_{1}} \\ o_{2} G_{o_{2}} \\ o_{3} G_{o_{3}} \end{matrix}]

where

G_{o_{i}}

is the generator matrix for the linear code

{C_{o}}_{i}, for i = {1, 2, 3} .

Using the generator matrices above, we can say that the generator matrixG for the linear code over

R

is:

G = [\begin{matrix} G_{F_{q}} & 0 & 0 \\ 0 & G_{R_{1}} & 0 \\ 0 & 0 & G_{R_{2}} \end{matrix}] .

Note that the minimum distance of

C

is min

{d_{H} (C_{1}), d_{H} (φ_{1} (C_{e})), d_{H} (φ_{2} (C_{o}))} .

The following theorem provides the weight preserving nature of the Gray map.

Theorem 1.

The Gray map φ defined above is linear and weight preserving.

Proof.

Let

x = (x_{1}, x_{2}, x_{3}), x^{'} = (x_{1}^{'}, x_{2}^{'}, x_{3}^{'}) be in R_{γ}

where

x_{1}, x_{1}^{'} \in F_{q}^{n_{1}}, x_{2}, x_{2}^{'} \in R_{1}^{n_{2}}, x_{3},

x_{3}^{'} \in R_{2}^{n_{3}} .

We have:

\begin{matrix} φ (x + x^{'}) & = & φ (x_{1} + x_{1}^{'}, x_{2} + x_{2}^{'}, x_{3} + x_{3}^{'}) \\ = & (x_{1} + x_{1}^{'}, φ_{1} (x_{2} + x_{2}^{'}), φ_{2} (x_{3} + x_{3}^{'})) \\ = & (x_{1}, φ_{1} (x_{2}), φ_{2} (x_{3})) + (x_{1}^{'}, φ_{1} (x_{2}^{'}), φ_{2} (x_{3}^{'})) (∵ φ_{1} and φ_{2} are linear) \\ = & φ (x) + φ (x^{'}) \end{matrix}

Using the linear map

φ,

d_{G} (x, x^{'}) = w t_{G} (x - x^{'}) = w t_{H} (φ (x) - φ (x^{'})) = d_{H} (φ (x), φ (x^{'}))

Hence,

φ

is a weight preserving linear map. □

The following theorem gives the parameters of the Gray image of a linear code.

Theorem 2.

C \subseteq R_{γ}

is an

(n_{1} + n_{2} + n_{3}, d_{G})

linear code then

φ (C)

is an

(n_{1} + 2 (n_{2}) + 3 (n_{3}), d_{H})

linear code over

F_{q},

where

d_{G} = d_{H} .

Proof.

The proof can extended from the proof of Theorem 1. □

The following Theorem characterizes

φ (C)

Theorem 3.

C \subseteq R_{γ}

is linear, then

φ (C) = C_{1} \otimes_{i = 1}^{i = 2} C_{e_{i}} \otimes_{j = 1}^{j = 3} C_{o_{j}}, ∣ C ∣ = ∣ C_{1} ∣ \prod_{i = 1}^{i = 2} ∣ C_{e_{i}} ∣ \prod_{j = 1}^{j = 3} ∣ C_{o_{j}} ∣

Proof.

Let:

φ (x, y, z) = (x, φ_{1} (y), φ_{3} (z)) = (a_{1}, \dots, a_{6}) \in φ (C) \subseteq F_{q}^{6} .

Note that

φ

is bijective and

C = C_{1} \otimes C_{e} \otimes C_{o}

is linear. Thus,

a_{1} = x \in C_{1} . Also note that φ_{1} (y) = (a_{2}, a_{3}), φ_{2} (z) = (a_{4}, a_{5}, a_{6}) . Sin ce φ_{i}^{'} s

are bijective,

a_{2} e_{1} + a_{3} e_{2} \in e_{1} C_{e_{1}} \oplus e_{2} C_{e_{2}} = C_{e} .

Hence,

(a_{2}, a_{3}) \in C_{e_{1}} \otimes C_{e_{2}}

and similarly

(a_{4}, a_{5}, a_{6}) \in C_{o_{1}} \otimes C_{o_{2}} \otimes C_{o_{2}} .

The converse holds in a similar way. The second part of the statement follows from the fact that

φ

is bijective. □

The following Theorem furnishes the decompostion of the dual of the linear code

C .

Theorem 4.

C = C_{1} \otimes C_{e} \otimes C_{o}

is a linear code over

R then C^{⊥} = C_{1}^{⊥} \otimes C_{e}^{⊥} \otimes C_{o}^{⊥}, where C_{1}^{⊥},

C_{e}^{⊥}

and

C_{o}^{⊥}

are duals for the respective linear codes.

Proof.

Let

C^{⊥} = {c^{'} \in R_{γ} | 〈 c, c^{'} 〉 = 0 for all c \in C} = {(x^{'}, y^{'}, z^{'}) \in R_{⚪} | x^{'} \in F_{q}^{n_{1}},

y^{'} \in R_{1}^{n_{2}}, z^{'} \in R_{2}^{n_{3}}} .

Let

c = (x, y, z) \in C = C_{1} \otimes C_{e} \otimes C_{o} .

Then:

〈 c, c^{'} 〉 = x x^{'} + y y^{'} + z z^{'} = 0 .

Thus,

x^{'} \in C_{1}^{⊥}, y^{'} \in C_{e}^{⊥}, z^{'} \in C_{o}^{⊥} and so C^{⊥} \subseteq C_{1}^{⊥} \otimes C_{e}^{⊥} \otimes C_{o}^{⊥} . Since | C^{⊥} | = | C_{1}^{⊥} | | C_{e}^{⊥} | | C_{o}^{⊥} |,

the statement holds.

□

The next result shows that the Gray maps is compatible with duality.

Theorem 5.

C \subseteq R_{γ}

is linear, then

φ (C^{⊥}) = φ {(C)}^{⊥} .

Proof.

Let

(x, y, z) \in C

and

(x^{'}, y^{'}, z^{'}) \in C^{⊥}

, where

x \in C_{1}, y \in C_{e}, z \in C_{o}

and

x^{'} \in C_{1}^{⊥}, y^{'} \in C_{e}^{⊥}, z^{'} \in C_{o}^{⊥},

then

〈 (x | y | z, x^{'} | y^{'} | z^{'}) 〉 = 0 .

Using Theorem 4,

C_{1}^{⊥}, C_{e}^{⊥} and C_{o}^{⊥}

are duals for

C_{1}, C_{e}, and C_{o} .

Now, we have

φ (x, y, z) = (x, φ_{1} (y), φ_{2} (z)), φ (x^{'}, y^{'}, z^{'}) = (x^{'}, φ_{1} (y^{'}), φ_{2} (z^{'})),

then the inner product is given by:

\begin{matrix} 〈 φ (x, y, z), φ (x^{'}, y^{'}, z^{'}) 〉 & = & 〈 (x, φ_{1} (y), φ_{2} (z)), (x^{'}, φ_{1} (y^{'}), φ_{2} (z^{'})) 〉 \\ = & 〈 (x, 0, 0), (x^{'}, 0, 0) 〉 + 〈 (0, φ_{1} (y), 0), (0, φ_{1} (y^{'}), 0) 〉 \\ + 〈 (0, 0, φ_{2} (z)), (0, 0, φ_{2} (z^{'})) 〉 \\ = & 0 (∵ φ_{1} (C^{⊥}) = φ_{1} {(C)}^{⊥}, φ_{2} (C^{⊥}) = φ_{2} {(C)}^{⊥}) \end{matrix}

Thus,

φ (C^{⊥}) \subseteq φ {(C)}^{⊥} .

Since the cardinality is the same on both sides, the statement holds. □

The following result provides the self duality nature of the linear code and its Gray image.

Corollary 1.

C

is a linear

R_{γ}

-code, then

C

is self-dual iff

φ (C)

is self-dual. Moreover,

φ (C)

is a self-orthogonal code over

F_{q}

iff C is self-orthogonal.

Proof.

Let

C

be a self-dual linear code of lengthn over

R

. Thus,

C = C^{⊥}

. Then,

φ (C) = φ (C^{⊥})

, and hence, by Theorem 5, we have

φ (C) = {(φ (C))}^{⊥}

. Thus,

φ (C)

is a self-dual linear code of length

n_{1} + 2 n_{2} + 3 n_{3}

over

F_{q}

. Conversely, let

φ (C)

be a self-dual linear code of length

n_{1} + 2 n_{2} + 3 n_{3}

over

F_{q}

. Then,

φ (C) = {(φ (C))}^{⊥}

, and hence, by Theorem 5, we have

φ (C) = φ (C^{⊥})

. Since

φ

is bijection,

C = C^{⊥}

. Therefore,

C

is a self-dual linear code over

R_{γ}

. Similarly, the self orthogonal case holds. □

4. Skew Cyclic $R$ -Codes

Let

θ_{t}

be a non-trivial Frobenius automorphism defined by:

θ_{t} : F_{q} \to F_{q}, θ_{t} (a) = a^{p^{t}},

wheret divides

m .

It can be extended to

R_{1}

and

R_{2}

by:

θ_{t} (a + u b) = θ_{t} (a) + u θ_{t} (b), θ_{t} (a + u b + v c) = θ_{t} (a) + u θ_{t} (b) + v θ_{t} (b) .

Since

t | m,

the order of automorphism

θ_{t} is \frac{m}{t} .

We define a polynomial ring

R_{i} [x, θ_{t}]

(1 \leq i \leq 2)

as follows:

R_{i} [x, θ_{t}] = {a_{1} + \dots + a_{n} x^{n} | a_{j} \in R_{i}, 1 \leq j \leq n}

Clearly,

R_{i} [x, θ_{t}]

is a ring with respect to usual addition and the multiplication defined by:

a x^{m} b x^{n} = a θ_{t}^{m} (b) x^{m + n}

Note that it is a non-commutative ring unless

θ_{t}

is an identity map. A non-empty set

C

is said to be a linear code of length

n_{i}

over

R_{i}

if it is a

R_{i}

submodule of

R_{i}^{n_{i}} .

Using the above polynomial rings above, we extend the polynomial ring to

R

by:

R [x, θ_{t}] = {(a (x), b (x), c (x)) : a (x) \in F_{q} [x], b (x) \in R_{1} [x], c (x) \in R_{2} [x]} .

It can be seen that

R [x, θ_{t}]

is a

R_{2} [x; θ_{t}]

submodule with respect to usual addition and multiplication defined by:

\begin{matrix} ★ : R_{2} [x] \times R [x, θ_{t}] & \to & R [x, θ_{t}] \\ (a x^{s}) ★ (b_{1} x^{i}, b_{2} x^{j}, b_{3} x^{k}) & = & (η (a) x^{s} b_{1} x^{i}, δ (a) x^{s} b_{2} x^{j}, a x^{s} b_{3} x^{k}) \\ = & (η (a) θ_{t}^{s} (b_{1}) x^{s + i}, δ (a) θ_{t}^{s} (b_{2}) x^{s + j}, a θ_{t}^{s} (b_{3}) x^{s + k}) \end{matrix}

However, under associative and distributive laws, the multiplication can be extended to

R_{γ} [x; θ_{t}] = \frac{F_{q} [x; θ_{t}]}{〈 x^{n_{1}} - 1 〉} \times \frac{R_{1} [x; θ_{t}]}{〈 x^{n_{2}} - 1 〉} \times \frac{R_{2} [x; θ_{t}]}{〈 x^{n_{3}} - 1 〉}

as follows:

★ : R_{2} [x; θ_{t}] \times R_{γ} [x; θ_{t}] \to R_{γ} [x; θ_{t}]

r (x) ★ (f_{1} (x) + 〈 x^{n_{1}} - 1 〉, f_{2} (x) + 〈 x^{n_{2}} - 1 〉, f_{3} (x) + 〈 x^{n_{3}} - 1 〉) = (η (r (x)) f_{1} (x) + 〈 x^{n_{1}} - 1 〉, δ (r (x)) f_{2} (x) + 〈 x^{n_{2}} - 1 〉,

r (x) f_{3} (x) + 〈 x^{n_{3}} - 1 〉) .

Definition 1

([2]).We say that an

R

-submodule

C

R^{n}

is a

θ_{t}

-cyclic code if for any

c = (c_{0}, c_{1}, \dots, c_{n - 1}) \in C

σ_{1} (c) = (θ_{t} (c_{n - 1}), θ_{t} (c_{0}), \dots, θ_{t} (c_{n - 2})) \in C

. The operator

σ_{1}

is then said to be a

θ_{t}

-cyclic shift operator on

R^{n}

Definition 2.

A non-trivial

R_{2}

-submodule

C

R_{γ}

is called a

θ_{t}

-cyclic code if for any

c = (c_{0, 1}, c_{1, 1}, \dots, c_{n_{1} - 1, 1}, c_{0, e}, c_{1, e}, \dots, c_{n_{2} - 1, e}, c_{0, o}, c_{1, o}, \dots, c_{n_{3} - 1, o}) \in C

σ (c) = (θ (c_{n_{1} - 1, 1}), θ (c_{0, 1}), \dots, θ (c_{n_{1} - 2, 1}), θ (c_{n_{2} - 1, e}), θ (c_{0, e}), \dots, θ (c_{n_{2} - 2, e}), θ (c_{n_{3} - 1, o}), θ (c_{0, o}), \dots, θ (c_{n_{3} - 2, o})) \in C

. The operator σ is called a

θ_{t}

-cyclic shift operator on

R^{n}

The following result yields the relationship between the

θ_{t}

-cyclic codes over

R and F_{q}

Theorem 6.

Let

C = C_{1} \otimes C_{e} \otimes C_{o} \subseteq R_{γ}

be linear. Then

C

is a

θ_{t}

-cyclic code if and only if

C_{1}, C_{e}

and

C_{o}

are

θ_{t}

-cyclic codes of length

n_{1}, n_{2}

and

n_{3}

over

F_{q}, R_{1}

and

R_{2}

respectively.

Proof.

Let

C = C_{1} \otimes C_{e} \otimes C_{o}

be a

θ_{t}

-cyclic code over

R

. Let

z = (z_{1}, z_{e}, z_{o}) \in C,

that is:

\begin{matrix} z = (z_{0, 1}, z_{1, 1}, \dots, z_{n_{1} - 1, 1}, z_{0, e}, z_{1, e}, \dots, z_{n_{2} - 1, e}, z_{0, o}, z_{1, o}, \dots, z_{n_{3} - 1, o}) \in C . \end{matrix}

Then,

σ (z) = (θ (z_{n_{1} - 1, 1}), θ (z_{0, 1}), \dots, θ (z_{n_{1} - 2, 1}), θ (z_{n_{2} - 1, e}), θ (z_{0, e}), \dots, θ (z_{n_{2} - 2, e}),

θ (z_{n_{3} - 1, o}), θ (z_{0, o}), \dots, θ (z_{n_{3} - 2, o})) = (σ (z_{1}), σ (z_{e}), σ (z_{o})) \in C

. From this, we can conclude that:

\begin{matrix} σ (z_{1}) \in C_{1}, σ (z_{e}) \in C_{e} and σ (z_{o}) \in C_{o} \end{matrix}

Hence,

C_{1}, C_{e}

, and

C_{o}

are

θ_{t}

-cyclic code of length

n_{i} .

The converse holds in a similar way.

□

We recall the following Theorem from [9].

Theorem 7.

([9]).Let

C_{o} = o_{1} C_{o_{1}} \oplus o_{2} C_{o_{2}} \oplus o_{3} C_{o_{3}}

be a linear code over

R_{2}

of length

n_{3}

, then

C_{o}

θ_{t}

-cyclic code iff

C_{o_{i}} (1 \leq i \leq 3)

is a

θ_{t}

-cyclic code of length

n_{3}

over

F_{q} .

The analogue of this result in our setting is as follows.

Theorem 8.

([8]).Let

C_{e} = e_{1} C_{e_{1}} \oplus e_{2} C_{e_{2}}

be a linear code over

R_{1}

of length

n_{2}

then

C_{e}

θ_{t}

-cyclic code iff

C_{e_{i}} (1 \leq i \leq 2)

is a

θ_{t}

-cyclic code of length

n_{2}

over

F_{q} .

Theorem 9.

C = C_{1} \otimes C_{e} \otimes C_{o}

is a linear code of length

γ = n_{1} + n_{2} + n_{3}

, then

C

θ_{t}

-cyclic iff

C_{1}, C_{e_{i}}, C_{o_{j}} (1 \leq i \leq 2, 1 \leq j \leq 3)

are

θ_{t}

-cyclic code of length

n_{1}, n_{2}, n_{3}

over

F_{q}

respecively.

Proof.

We obtain the proof on combining proofs of Theorems 6–8. □

These notions are well-behaved with respect to duality as the next result shows.

Theorem 10.

C

is a

θ_{t}

-cyclic code of length

n,

then its dual

C^{⊥}

is also a

θ_{t}

-cyclic code.

Proof.

From Theorem 9,

C_{1}, C_{e_{i}}, C_{o_{j}} (1 \leq i \leq 2, 1 \leq j \leq 3)

are

θ_{t}

-cyclic codes over

F_{q} .

Then,

C_{1}^{⊥}, C_{e_{i}}^{⊥}, C_{o_{j}}^{⊥} (1 \leq i \leq 2, 1 \leq j \leq 3)

are

θ_{t}

-cyclic codes over

F_{q}

from [29] and once again by using Theorem 9,

C^{⊥}

becomes a

θ_{t}

-cylic code. □

Recall the following result from [4].

Lemma 1

([4]).Let C be a

θ_{t}

-cyclic code of length n over

F_{q}

. Then, there exists a polynomial

f (x) \in F_{q} [x; θ_{t}]

such that

C = 〈 f (x) 〉

and

x^{n} - 1 = g (x) f (x)

F_{q} [x; θ_{t}]

By assuming

o (θ_{t}) | n,

the counterpart follows.

Theorem 11.

Let

C = C_{1} \otimes C_{e} \otimes C_{o}

be a

θ_{t}

-cyclic code of length n over

R

and assume that the order of

θ_{t}

divides

n .

Then,

C = 〈 B_{1}, B_{e}, B_{o} 〉

, where

B_{1} = 〈 (f_{1} (x), 0, 0) 〉, B_{e} = 〈 (0, f_{e} (x), 0) 〉,

and

B_{o} = 〈 (b_{1} (x), b_{e} (x), f_{o} (x)) 〉,

such that

C_{1} = 〈 f_{1} (x) 〉, C_{e} = 〈 f_{e} (x) 〉, C_{o} = 〈 f_{o} (x) 〉, b_{1} (x) \in C_{1}

and b_{2} (x) \in C_{e} .

Proof.

Let

C = C_{1} \otimes C_{e} \otimes C_{o}

be a

θ_{t}

-cyclic code of length

γ = n_{1} + n_{2} + n_{3}

over

R

. Then, by Thereom 6,

C_{1}, C_{e}, C_{o}

are

θ_{t}

-cyclic codes of length

n_{i}

over

F_{q}, R_{1}

and

R_{2}

. Define a homomorphism from

C

R

as follows:

\begin{matrix} ψ : C & \mapsto & R \\ ψ (c_{1} (x), c_{e} (x), c_{o} (x)) & = & (0, 0, c_{o} (x)) \end{matrix}

Define:

k e r (ψ) = {(c_{1} (x), c_{e} (x), 0) : c_{1} (x) \in C_{1}, c_{e} (x) \in C_{e}}

I = {(c_{1} (x), c_{e} (x)) \in F_{q} [x; θ_{t}] \times R_{1} [x; θ_{t}] : (c_{1} (x), c_{e} (x), 0) \in \ker (ψ)} .

Clearly,

I = I_{1} \times I_{e}

forms a submodule of

F_{q} [x; θ_{t}] \times R_{1} [x; θ_{t}] .

Therefore, there exist a polynomial

f_{1} (x)

and

f_{e} (x)

F_{q} [x; θ_{t}]

and

R_{1} [x; θ_{t}]

, respectively, generating

I_{1}

and

I_{e}

with

f_{1} (x) | x^{n_{1}} - 1

and

f_{e} (x) | x^{n_{e}} - 1 .

Thus,

I = 〈 (f_{1} (x), 0), (0, f_{e} (x)) 〉,

then for any

(c_{1} (x), c_{e} (x), 0) \in \ker (ψ),

(c_{1} (x), c_{e} (x)) = v (x) ★ (f_{1} (x), 0), (0, f_{e} (x))

for some

v (x) \in R_{1} [x; θ_{t}] .

Finally, it leads to ker

(ψ) = 〈 (f_{1} (x), 0, 0), (0, f_{e} (x), 0) 〉 .

The fact that

C

is a submodule implies that

ψ (C)

is a submodule. By using the first isomorphism theorem:

\begin{matrix} C / k e r (ψ) ≅ ψ (C) . \end{matrix}

Let

(b_{1} (x), b_{e} (x), f_{o} (x)) \in C

, then

ψ (b_{1} (x), b_{e} (x), f_{o} (x)) = (0, 0, f_{o} (x)) .

From this, any

θ_{t}

-cyclic code of lengthn can be represented by

C = 〈 (f_{1} (x), 0, 0) (0, f_{e} (x), 0), (b_{1} (x), b_{e} (x), f_{o} (x)) 〉

, where

f_{1} (x) | (x^{n_{1}} - 1), f_{e} (x) | (x^{n_{2}} - 1)

and

f_{o} (x) | (x^{n_{3}} - 1) .

□

Furthermore, we have

C

θ_{t}

-cyclic, then

C_{k}

, where

k \in {1, e_{1}, e_{2}, o_{1}, o_{2}, o_{3}}

is skew

θ_{t}

-cyclic code over

F_{q}

with respective lengths. From Theorem 3,

| C | = ∣ C_{1} ∣ \prod_{i = 1}^{i = 2} ∣ C_{e_{i}} ∣ \prod_{j = 1}^{j = 3} ∣ C_{o_{j}} ∣

, since each

C_{k}

θ_{t}

-cyclic it is generated by a polynomial

f_{k} (x)

, and thus,

| C | = q^{γ - \sum_{i = 1}^{6} ϵ_{k}}

, where

γ = n_{1} + 2 (n_{2}) + 3 (n_{3}) .

The following Theorem provides the generator polynomials for

θ_{t}

-cylic codes over

F_{q} .

Theorem 12.

Let

C = C_{1} \otimes C_{e} \otimes C_{o}

be a skew cyclic code over

R

of length

γ = n_{1} + n_{2} + n_{3}

. Then, there exists a polynomial:

(i): $f_{1} (x) \in F_{q} [x; θ_{t}]$ such that $C_{1} = 〈 f_{1} (x) 〉$ and $x^{n_{1}} - 1 = g_{1} (x) f_{1} (x)$ .
(ii): $f_{e} (x) \in R_{1} [x; θ_{t}]$ such that $C_{e} = 〈 f_{e} (x) 〉$ and $x^{n_{2}} - 1 = g_{e} (x) f_{e} (x) where f_{e} (x) = \sum_{i = 1}^{2} e_{i} f_{e_{i}} (x)$ .
(iii): $f_{o} (x) \in R_{2} [x; θ_{t}]$ such that $C_{o} = 〈 f_{o} (x) 〉$ and $x^{n_{3}} - 1 = g_{o} (x) f_{o} (x) where f_{o} (x) = \sum_{i = 1}^{3} o_{i} f_{o_{i}} (x)$ .

Proof.

Let

C

be a

θ_{t}

-cyclic code of length

γ = n_{1} + n_{2} + n_{3} .

From Theorem 6, we have that

C_{1}, C_{e}

, and

C_{o}

are

θ_{t}

-cyclic codes. Using Lemma 1,

(i)

follows.

Then, the proof of

(i i)

is as follows. Let

C_{e} = e_{1} C_{e_{1}} \oplus e_{2} C_{e_{2}}

be a

θ_{t}

-cyclic code of length

n_{2}

over

R_{1} .

Thereom 7 says that,

C_{e_{1}}

and

C_{e_{2}}

are

θ_{t}

-cyclic codes of length

n_{2}

over

F_{q}

. Lemma 1 says that we have

C_{i} = 〈 f_{e_{i}} (x) 〉

and

x^{n_{2}} - 1 = g_{e_{i}} (x) f_{e_{i}} (x)

F_{q} [x; θ_{t}]

for

i \in {1, 2}

. Then,

e_{i} f_{e_{i}} (x) \in C

for

i \in {1, 2}

. Also, for any

f_{e} (x) \in C

, we have

f_{e} (x) = \sum_{i = 1}^{2} e_{i} h_{e_{i}} (x) f_{e_{i}} (x)

, where

h_{e_{i}} (x) \in F_{q} [x; θ_{t}]

for

i \in {1, 2}

. Thus,

f_{e} (x) \in 〈 e_{1} f_{e_{1}} (x), e_{2} f_{e_{2}} (x) 〉

. Therefore,

C = 〈 e_{1} f_{e_{1}} (x), e_{2} f_{e_{2}} (x) 〉

. As

x^{n_{2}} - 1 = g_{e_{i}} (x) f_{e_{i}} (x)

F_{q} [x; θ_{t}]

for

i \in {1, 2} .

Let

f_{e} (x) = e_{1} f_{e_{1}} (x) + e_{2} f_{e_{2}} (x) \in R_{1} [x; θ_{t}]

. Then,

f_{e} (x) \in C

. On the other hand

e_{i} f_{e_{i}} (x) = e_{i} f_{e} (x) \in 〈 f_{e} (x) 〉

for

i = 1, 2

. Consequently,

C = 〈 f_{e} (x) 〉

. Furthermore,

[\sum_{i = 1}^{2} e_{i} g_{e_{i}} (x)] f_{e} (x) = \sum_{i = 1}^{2} e_{i} g_{e_{i}} (x) f_{e_{i}} (x) = \sum_{i = 1}^{2} e_{i} (x^{n_{2}} - 1) = x^{n_{2}} - 1

. Then,

x^{n_{2}} - 1 = g_{e} (x) f_{e} (x)

R_{1} [x; θ_{t}]

, where

g_{e} (x) = \sum_{i = 1}^{2} e_{i} g_{e_{i}} (x)

. Thus,

(i i)

follows.

(i i i)

is similar to the proof of

(i i) .

□

5. Skew Constacyclic Code over $R$

In this section, we study skew

θ_{t}

-constacyclic codes over

R

. We choose a unit element

α \in R_{2}^{*} such that α

satisfies the condition

α^{2} = 1, (α = 1, - 1, \dots) .

Definition 3.

Let

α_{i} \in F_{p^{t}} ∖ {0} .

A linear code

C \subseteq R_{γ} [x, θ]

is called skew

α = α_{1} + u α_{2} + v α_{3}

-constacyclic code if it is invariant under the cyclic shift operator

λ_{α}

, which is whenever:

\begin{matrix} c & = & (x_{0}, x_{1}, \dots, x_{n_{1} - 1}, y_{0}, y_{1}, \dots, y_{n_{2} - 1}, z_{0}, z_{1}, \dots, z_{n_{3} - 1}) \in C \\ λ_{α} (c) & = & (α_{1} θ_{t} (x_{n_{1} - 1}), θ_{t} (x_{0}), \dots, θ_{t} (x_{n_{1}}), (α_{1} + u α_{2}) θ_{t} (y_{n_{1} - 1}), θ_{t} (y_{1}), \dots, θ_{t} (y_{n_{2} - 2}), \\ (α_{1} + u α_{2} + v α_{3}) θ_{t} (z_{n_{3} - 1}), θ_{t} (z_{0}), \dots, θ_{t} (z_{n_{3} - 2})) \in C \end{matrix}

The following two results translate symmetry conditions into algebraic constraints. We give the first result without proof.

Theorem 13.

Let

R_{n, λ} = R [x; θ_{t}] / 〈 x^{n} - λ 〉

. A linear code C of length n over R is

(θ_{t}, λ)

-cyclic code if and only if C is a left

R [x; θ_{t}]

-submodule of

R_{n, λ}

The second result is less immediate.

Theorem 14.

A code

C

is skew α-cyclic code over

R_{γ} = \frac{F_{q} [x, θ_{t}]}{x^{n_{1}} - α} \times \frac{R_{1} [x, θ_{t}]}{x^{n_{2}} - α} \times \frac{R_{2} [x, θ_{t}]}{x^{n_{3}} - α}

iff

C

is a left

R_{2} [x, θ_{t}]

module over

R_{γ} .

Proof.

Let

C

be a skew

α

-cyclic code. Then, by definition

x ★ (f (x) | g (x) | h (x)) \in C

\begin{matrix} x ★ (f (x) | g (x) | h (x)) = (θ_{t} (f_{0}) x + θ_{t} (f_{1}) x^{2} + \dots + α_{1} θ_{t} (f_{n_{1} - 1}), θ_{t} (g_{0}) x + θ_{t} (g_{1}) x^{2} + \dots + (α_{1} + u α_{2}) θ_{t} (g_{n_{2} - 1}), \\ θ_{t} (h_{0}) x + θ_{t} (h_{1}) x^{2} + \dots + (α_{1} + u α_{2} + v α_{3}) θ_{t} (h_{n_{3} - 1})) \in C \end{matrix}

Moreover, by using linearity of

C

r (x) ★ (g_{1} (x) | g_{2} (x) | g_{3} (x)) \in C

for some

r (x) \in R_{2} [x, θ_{t}] .

Hence,

C

is an left

R_{2} [x, θ_{t}]

submodule over

R_{γ} .

Conversely, assume that

C

is an left

R_{2} [x, θ_{t}]

submodule over

R_{γ},

then we have

x ★ (f (x) | g (x) | h (x)) \in C

implies

C

is skew

α

-cyclic code. □

Theorem 15.

The code

C_{o} \subseteq R_{2}^{n}

is skew

α = α_{1} + u α_{2} + v α_{3}

-cyclic of length n iff

C_{o_{1}}, C_{o_{2}}

, and

C_{o_{3}}

are skew

α_{1}, α_{1} + α_{2}, α_{1} + α_{3}

-cyclic codes over

F_{q}

of length

n .

Proof.

Let

C_{o}

be a skew

α

-cyclic code. Let

a = x o_{1} + y o_{2} + z o_{3} \in C_{o}

, where

x = (x_{0}, x_{1}, \dots, x_{n - 1}) \in C_{o_{1}}, y = (y_{0}, y_{1}, \dots, y_{n - 1}) \in C_{o_{2}}

and

x = (z_{0}, z_{1}, \dots, z_{n - 1}) \in C_{o_{3}} .

Then, we have by definiton,

λ_{α} (x) \in C_{o},

\begin{matrix} λ_{α} (o_{1} (x_{0}, x_{1}, \dots, x_{n - 1}) + o_{2} (y_{0}, y_{1}, \dots + y_{n - 1}) \\ + o_{3} (z_{0}, z_{1}, \dots, z_{n - 1})) & = & ((α_{1} + u α_{2} + v α_{3}) ★ o_{1} (θ_{t} (x_{n - 1}), θ_{t} (x_{0}), \dots, θ_{t} (x_{n - 2})) + \\ (α_{1} + u α_{2} + v α_{3}) ★ o_{2} (θ_{t} (y_{n - 1}), θ_{t} (y_{0}), \dots + θ_{t} (y_{n - 2})) + \\ (α_{1} + u α_{2} + v α_{3}) ★ o_{3} (θ_{t} (z_{n - 1}), θ_{t} (z_{0}), \dots, θ_{t} (z_{n - 1}))) \\ \Rightarrow & λ_{α_{1}} (x) + λ_{α_{1} + α_{2}} (y) + λ_{α_{1} + α_{3}} (z) \in C_{o} \\ \Rightarrow & λ_{α_{1}} (x) \in C_{o_{1}}, λ_{α_{1} + α_{2}} (y) \in C_{o_{2}}, λ_{α_{1} + α_{3}} (z) \in C_{o_{3}} \end{matrix}

Hence,

C_{o_{1}}, C_{o_{2}}, and C_{o_{3}}

are skew

α_{1}, α_{1} + α_{2}, α_{1} + α_{3}

-cyclic codes over

F_{q}

of length

n .

Conversely, assume that

C_{o_{1}}, C_{o_{2}} and C_{o_{3}}

are skew

α_{1}, α_{1} + α_{2}, α_{1} + α_{3}

-cyclic codes over

F_{q}

of length

n .

Let

m_{0}, m_{1}, \dots, m_{n - 1}

be an element in

C_{o},

where

m_{i} = o_{1} x_{i} + o_{2} y_{i} + o_{3} z_{i}

such that

x = (x_{0}, x_{2}, \dots, x_{n - 1}) \in C_{o_{1}}, y = (y_{0}, y_{2}, \dots, y_{n - 1}) \in C_{o_{2}}

and

z = (z_{0}, z_{2}, \dots, z_{n - 1}) \in C_{o_{3}} .

Then we have

λ_{α_{1}} (x) \in C_{o_{1}}, λ_{α_{1} + α_{2}} (y) \in C_{o_{2}} and λ_{α_{1} + α_{3}} (z) \in C_{o_{3}} .

So we get,

\begin{matrix} o_{1} λ_{α_{1}} (x) + o_{2} λ_{α_{1} + α_{2}} (y) + o_{3} λ_{α_{1} + α_{3}} (z) & = & o_{1} λ_{α_{1}} (x_{0}, x_{1}, \dots, x_{n - 1}) + o_{2} λ_{α_{1} + α_{2}} (y_{0}, y_{1}, \dots + y_{n - 1}) \\ + o_{3} λ_{α_{1} + α_{3}} (z_{0}, z_{1}, \dots, z_{n - 1}) \in C \\ = & λ_{α} (m_{0}, m_{1}, \dots, m_{n - 1}) \in C \end{matrix}

Hence,

C

is skew

α -

cyclic code over

R_{2}^{n} .

□

Theorem 16.

C_{e}

be a a skew

α = α_{1} + u α_{2}

-cyclic code over

R_{1}

iff

C_{e_{1}}

and

C_{e_{2}}

are skew

α_{1} + α_{2}

and

α_{1}

-cyclic codes over

F_{q} .

Proof.

The proof is similar to Theorem 15 taking modv to the above condition. □

Theorem 17.

C

be a skew

α = α_{1} + u α_{2} + v α_{3}

-cyclic code over

R

of length

γ = n_{1} + n_{2} + n_{3}

iff

C_{1}, C_{e} and C_{o}

are

α_{1}, α_{1} + u α_{2}

, and

α_{1} + u α_{2} + v α_{3}

-cyclic codes over

F_{q}, R_{1} and R_{2}

, respectively.

Proof.

C_{1}, C_{e} and C_{o}

α_{1}, α_{1} + u α_{2}

, and

α_{1} + u α_{2} + v α_{3}

-cyclic. Consider

x = (x_{0}, x_{1}, \dots, x_{n_{1} - 1}),

y = (y_{0}, y_{1}, \dots, y_{n_{2} - 1})

and

z = (z_{0}, z_{1}, \dots, z_{n_{3} - 1}) .

Consider

α_{1} + u α_{2} = β .

Then, we have:

(x, y, z) \in C \Rightarrow (λ_{α_{1}} (x), λ_{β} (y), λ_{α} (y)) \in C

Hence,

C

is skew

α

-cyclic. The converse part holds similarly. □

Theorem 18.

C

be a skew α-cyclic code of length

γ = n_{1} + n_{2} + n_{3}

iff

C_{1}

is skew

α_{1}

-cyclic code of length

n_{1}

C_{e_{1}}

, and

C_{e_{2}}

are

α_{1} + α_{2}, α_{1}

-cyclic codes of length

n_{2}

and

C_{o_{1}}, C_{o_{2}}

, and

C_{o_{3}}

are skew

α_{1}, α_{1} + α_{2}, α_{1} + α_{3}

-cyclic codes over

F_{q}

of length

n_{3} .

Proof.

Using Theorems 15–17 the result follows. □

Theorem 19.

C

be a skew

α = α_{1} + u α_{2} + v α_{3}

-cyclic code over

R

of length

γ = n_{1} + n_{2} + n_{3}

iff

C_{1}^{⊥}, C_{e}^{⊥}, and C_{o}^{⊥}

are

{(α_{1})}^{- 1}, {(α_{1} + u α_{2})}^{- 1}

, and

{(α_{1} + u α_{2} + v α_{3})}^{- 1}

-cyclic.

Proof.

Let

C

be a skew

α

-cyclic code, Lemma 3.1 [5] says that

C^{⊥}

is skew

{(α_{1} + u α_{2} + v α_{3})}^{- 1}

-cyclic code. From Theorem 17, we have

C_{1}^{⊥}, C_{e}^{⊥} and C_{o}^{⊥}

are skew

{(α_{1})}^{- 1}, {(α_{1} + u α_{2})}^{- 1}

and

{(α_{1} + u α_{2} + v α_{3})}^{- 1}

-cyclic. □

Corollary 2.

Let

C = C_{1} \otimes C_{e} \otimes C_{o} .

be a skew

α = α_{1} + u α_{2} + v α_{3}

-cyclic code over

R

of length

γ = n_{1} + n_{2} + n_{3}

. Then, there exist polynomials:

(i): $f_{1} (x) \in F_{q} [x; θ_{t}]$ such that $C_{1} = 〈 f_{1} (x) 〉$ and $x^{n_{1}} - α_{1} = g_{1} (x) f_{1} (x)$ .
(ii): $f_{e} (x) \in R_{1} [x; θ_{t}]$ such that $C_{e} = 〈 f_{e} (x) 〉$ and $x^{n_{2}} - (α_{1} + u α_{2}) = g_{e} (x) f_{e} (x)$ .
(iii): $f_{o} (x) \in R_{2} [x; θ_{t}]$ such that $C_{o} = 〈 f_{o} (x) 〉$ and $x^{n_{3}} - (α_{1} + u α_{2} + v α_{3}) = g_{o} (x) f_{o} (x)$ .

Proof.

The proof is similar to the proof of Theorem 12. □

Theorem 20.

Let

C = C_{1} \otimes C_{e} \otimes C_{o}

be a

θ_{t}

-constacyclic code of length γ over

R

. Then

C = 〈 B_{1}, B_{2}, B_{o} 〉

, where

B_{1} = 〈 (f_{1} (x), 0, 0) 〉, B_{1} = 〈 (0, f_{e} (x), 0) 〉,

and

B_{1} = 〈 (b_{1} (x), b_{e} (x), f_{o} (x)) 〉

Proof.

The proof is similar to the proof of Theorem 11. □

Example 1.

Let

q = 9

and

F_{9} = F_{3} [z]

with

z^{2} + 1 = 0 .

Consider the ring

R_{γ} = \frac{F_{9} [x, θ_{3}]}{x^{4} - 1} \times \frac{R_{1} [x, θ_{3}]}{x^{5} - 1} \times \frac{R_{2} [x, θ_{3}]}{x^{5} - 1}

, where

θ_{3}

is the Frobenius automorphism defined by

θ_{3} (a) = a^{3}

for any

a \in F_{9} .

Write:

x^{4} - 1 = (x + 1) (x + 2) (x + z) (x + 2 z) \in F_{9} [x, θ_{3}]

x^{5} - 1 = (x + 2) (x^{4} + x^{3} + x^{2} + x + 1) \in F_{9} [x, θ_{3}]

f_{1} (x) = 〈 (x + 1) 〉, f_{e} (x) = 〈 e_{1} (x + 2) + e_{2} (x + 2) 〉, f_{o} (x) = 〈 o_{1} (x + 2) + o_{2} (x + 2) + o_{3} (x + 2) 〉

By Theorem 12, we have that

f_{i}

divides

x^{n_{i}} - 1

for

(i = 1, e, o)

, yielding a code with parameter

[29, 18, 2]

over

F_{9} .

Example 2.

Let

q = 25

and

F_{25} = F_{5} [z]

with

z^{2} + z + 1 = 0 .

Consider the ring

R_{γ} = \frac{F_{25} [x, θ_{5}]}{x^{4} - 1} \times \frac{R_{1} [x, θ_{5}]}{x^{6} - 1} \times \frac{R_{2} [x, θ_{5}]}{x^{4} - 1}

, where

θ_{5}

is the Frobenius automorphism defined by

θ_{5} (a) = a^{5}

for any

a \in F_{25} .

Write

x^{4} - 1 = (x + 2) (x + 3) (x + z) (x + z + 1) \in F_{25} [x, θ_{5}]

x^{6} - 1 = (x^{2} - 1) (x^{2} + x + 1) (x^{2} - x + 1) \in F_{25} [x, θ_{5}]

f_{1} (x) = 〈 (x + 2) 〉, f_{e} (x) = 〈 e_{1} (x^{2} - 1) + e_{2} (x^{2} - 1) 〉, f_{o} (x) = 〈 o_{1} (x + z + 1) + o_{2} (x + z + 1) + o_{3} (x + z + 1) 〉

By Theorem 12, we have that

f_{i}

divides

x^{n_{i}} - 1

for

(i = 1, e, o)

yielding a code with parameter

[28, 20, 2]

over

F_{25} .

6. Conclusions and Open Problems

In this note, we have studied the algebraic and metric structure of skew cyclic and skew constacyclic codes over a special mixed alphabet. Thus, our codes have a structure of module over the largest of the three alphabets

R_{2} .

Codes over the product ring

F_{q} \times R_{1} \times R_{2}

would be modules over that larger ring. The two algebraic structures are different and should not be confused.

The present work leads itself to two paths of generalization: consider different mixed alphabets or replace the concepts of cyclicity by that of quasi-cyclicity. The former path seems easier than the latter, in view of the many examples of rings that have been used as alphabets of cyclic codes in recent years. On the other hand, the structure of quasi-cyclic codes is always more subtle than that of cyclic codes.

Author Contributions

Conceptualization, K.G., C.D., S.Ç. and P.S.; Methodology, C.M. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Data Availability Statement

No new data were created or analyzed in this study. Data sharing is not applicable to this article.

Conflicts of Interest

The authors declare no conflicts of interest.

References

Berlekamp, E.R. Negacyclic codes for the Lee metric. InCombinatorial Mathematics; Bose, R.C., Dowling, T.A., Eds.; University of North Carolian Press: Chapel Hill, NC, USA, 1969; pp. 298–316. [Google Scholar]
Boucher, D.; Geiselmann, W.; Ulmer, F. Skew cyclic codes.Appl. Algebra Eng. Comm.2007,18, 379–389. [Google Scholar] [CrossRef]
Boucher, D.; Sol, P.; Ulmer, F. Skew constacyclic codes over Galois rings.Adv. Math. Commun.2011,2, 273–292. [Google Scholar] [CrossRef]
Siap, I.; Abualrub, T.; Aydin, N.; Seneviratne, P. Skew cyclic codes of arbitrary length.Int. J. Inf. Coding Theory2011,2, 10–20. [Google Scholar] [CrossRef]
Jitman, S.; Ling, S.; Udomkavanich, P. Skew constacyclic codes over finite chain ring.Adv. Math. Commun.2012,6, 39–63. [Google Scholar]
Abualrub, T.; Aydin, N.; Seneviratne, P. Onθ-cyclic codes overF₂+vF₂.Aust. J. Comb.2012,54, 115–126. [Google Scholar]
Ashraf, M.; Mohammad, G. Skew cyclic codes $F_{3} + v F_{3}$ .Int. J. Inf. Coding Theory2014,24, 218–225. [Google Scholar]
Gursoy, F.; Siap, I.; Yildiz, B. Construction of skew cyclic codes over $F_{q} + v F_{q}$ .Adv. Math. Commun.2014,8, 313–322. [Google Scholar] [CrossRef]
Ashraf, M.; Mohammad, G. Skew cyclic codes $F_{q} + u F_{q} + v F_{q}$ .Asian-Eur. J. Math.2018,115, 1850072. [Google Scholar] [CrossRef]
Dertli, A.; Cengellenmis, Y. Skew cyclic codes over $F_{q} + u F_{q} + v F_{q} + u v F_{q}$ .J. Sci. Arts2017,2, 215–222. [Google Scholar]
Yao, T.; Shi, M.; Sole, P. On Skew cyclic codes over $F_{q} + u F_{q} + v F_{q} + u v F_{q}$ .J. Algebra Comb. Discret. Struct. Appl.2015,2, 163–168. [Google Scholar]
Gao, J.; Ma, F.; Fu, F. Skew constacyclic codes over the ring $F_{q} + v F_{q}$ .Appl. Comput. Math.2017,6, 286–295. [Google Scholar]
Islam, H.; Prakash, O. Skew cyclic and skew (α₁+uα₂+vα₃+uvα₄)-constacyclic codes over $F_{q} + u F_{q} + v F_{q} + u v F_{q}$ .Int. J. Inf. Coding Theory2018,5, 101–116. [Google Scholar]
Islam, H.; Prakash, O. A note on skew constacyclic codes over $F_{q} + u F_{q} + v F_{q}$ .Discret. Math. Algorithms Appl.2019,11, 1950030. [Google Scholar] [CrossRef]
Bhardwaj, S.; Raka, M. Skew constacyclic codes over a non-chain ring $F_{q}$ [u,v]/〈f(u),g(v),uv-vu〉.arXiv2019, arXiv:1905.12933. [Google Scholar]
Borges, J.; Fernandez-Cordoba, C.; Pujol, J.; Rifa, J. $z_{2} z_{4}$ -linear codes: Generator matrices and duality.Des. Codes Cryptogr.2010,54, 167–179. [Google Scholar] [CrossRef]
Fernandez-Cordoba, C.; Pujol, J.; Villanueva, M. $z_{z} z_{4}$ -linear codes: Rank and kernel.Des. Codes Cryptogr.2010,56, 43–59. [Google Scholar] [CrossRef]
Rifa-Pous, H.; Rifa, J.; Ronquillo, L. $z_{z} z_{4}$ -additive perfect codes in steganography.Adv. Math. Commun.2011,5, 425–433. [Google Scholar]
Aydogdu, I.; Siap, I. The structure of $Z_{2} Z_{2^{s}}$ -additive codes: Bounds on the minimum distance.Appl. Math. Inf. Sci.2013,7, 2271–2278. [Google Scholar] [CrossRef]
Aydogdu, I.; Siap, I. On $Z_{p^{r}} Z_{p^{s}}$ -additive codes.Linear Multilinear Algebra2015,63, 2089–2102. [Google Scholar] [CrossRef]
Shi, M.; Wu, R.; Krotov, D.S. On $Z_{p} Z_{p^{k}}$ -additive codes and their duality.IEEE Trans. Inf. Theory2018,65, 3842–3847. [Google Scholar]
Yao, T.; Zhu, S. $Z_{p} Z_{p^{s}}$ -additive cyclic codes are asymptotically good.Cryptogr. Commun.2020,12, 253–264. [Google Scholar] [CrossRef]
Yao, T.; Zhu, S.; Kai, X. Asymptotically good $Z_{p^{r}} Z_{p^{s}}$ -additive cyclic codes.Finite Fields Appl.2020,63, 101633. [Google Scholar] [CrossRef]
Abualrub, T.; Siap, I.; Aydin, N. $Z_{2} Z_{4}$ -Additive cyclic codes.IEEE Trans. Inform. Theory2014,60, 1508–1514. [Google Scholar] [CrossRef]
Borges, J.; Fernandez-Cordoba, C.; Ten-Valls, R. $Z_{2} Z_{4}$ -additive cyclic codes, generator polynomials and dual codes.IEEE Trans. Inform. Theory2016,62, 6348–6354. [Google Scholar] [CrossRef]
Aydogdu, I.; Abualrub, T.; Siap, I. On $Z_{2} Z_{2}$ [u]-additive codes.Int. J. Comput. Math.2015,92, 1806–1814. [Google Scholar] [CrossRef]
Aydogdu, I.; Abualrub, T.; Siap, I. The $Z_{2} Z_{2}$ [u]-cyclic and constacyclic codes.IEEE Trans. Inform. Theory2016,63, 4883–4893. [Google Scholar] [CrossRef]
Srinivasulu, B.; Maheshanand, B. The $Z_{2} (Z_{2} + u Z_{2})$ -additive cyclic codes and their duals.Discret. Math. Algorithms Appl.2016,8, 1793–8317. [Google Scholar] [CrossRef]
Boucher, D.; Ulmer, F. Coding with skew polynomial rings.J. Symb. Comput.2009,44, 1644–1656. [Google Scholar] [CrossRef]

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

© 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

Share and Cite

MDPI and ACS Style

Gowdhaman, K.; Mohan, C.; Durairajan, C.; Çalkavur, S.; Solé, P. Skew Cyclic and Skew Constacyclic Codes over a Mixed Alphabet.Axioms2024,13, 360. https://doi.org/10.3390/axioms13060360

AMA Style

Gowdhaman K, Mohan C, Durairajan C, Çalkavur S, Solé P. Skew Cyclic and Skew Constacyclic Codes over a Mixed Alphabet.Axioms. 2024; 13(6):360. https://doi.org/10.3390/axioms13060360

Chicago/Turabian Style

Gowdhaman, Karthick, Cruz Mohan, Chinnapillai Durairajan, Selda Çalkavur, and Patrick Solé. 2024. "Skew Cyclic and Skew Constacyclic Codes over a Mixed Alphabet"Axioms 13, no. 6: 360. https://doi.org/10.3390/axioms13060360

APA Style

Gowdhaman, K., Mohan, C., Durairajan, C., Çalkavur, S., & Solé, P. (2024). Skew Cyclic and Skew Constacyclic Codes over a Mixed Alphabet.Axioms,13(6), 360. https://doi.org/10.3390/axioms13060360

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further detailshere.

Article Metrics

Article Access Statistics

For more information on the journal statistics, clickhere.

Multiple requests from the same IP address are counted as one view.

Abstract

1. Introduction

2. Preliminaries

3. Decomposition and Properties of Gray Maps

4. Skew Cyclic $R$ -Codes

5. Skew Constacyclic Code over $R$

6. Conclusions and Open Problems

Author Contributions

Funding

Data Availability Statement

Conflicts of Interest

References

Share and Cite

Article Metrics

Article Access Statistics

Movatterモバイル変換

Journals

Topics

Information

Author Services

Initiatives

About

Notice

Notice

Article Menu

Need Help?

Support

Feedback

Information

JSmol Viewer

Skew Cyclic and Skew Constacyclic Codes over a Mixed Alphabet

Cite

Further Information

Guidelines

MDPI Initiatives

Follow MDPI

Share Link

Share

Movatterモバイル変換

Article Menu

Skew Cyclic and Skew Constacyclic Codes over a Mixed Alphabet

Abstract

1. Introduction

2. Preliminaries

3. Decomposition and Properties of Gray Maps

4. Skew CyclicR-Codes

5. Skew Constacyclic Code overR

6. Conclusions and Open Problems

Author Contributions

Funding

Data Availability Statement

Conflicts of Interest

References

Share and Cite

Article Metrics

Article Access Statistics

Further Information

Guidelines

MDPI Initiatives

Follow MDPI

4. Skew Cyclic $R$ -Codes

5. Skew Constacyclic Code over $R$