/compile/output.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 9, No. 3, 2016, 305-313

ISSN 1307-5543 – www.ejpam.com

On (1+ u)-Cyclic and Cyclic Codes over F2 + uF2 + vF2

Abdullah Dertli1,∗, Yasemin Cengellenmis2

1 Department of Mathematics, Faculty of Science and Arts, Ondokuz Mayıs University, Samsun,

Turkey
2 Department of Mathematics, Faculty of Science and Arts, Trakya University, Edirne, Turkey

Abstract. It is studied codes over the ring R= F2+uF2+vF2, u2 = 0, v2 = v,uv = vu= 0 which contains

the two ring F2+uF2,u2 = 0 and F2+ vF2, v2 = v. It is introduced (1+u)-cyclic codes and cyclic codes

over F2+uF2+ vF2. It is characterized codes over F2+ vF2 which are the images of (1+u)-cyclic codes

and cyclic codes over F2 + uF2+ vF2. It is obtained a representation of a linear code of length n over R

by means of C1 and C2 which are linear codes of length n over F2 + uF2. It is also characterized codes

over F2 which are the Gray images of (1+ u)-cyclic codes or cyclic codes over F2 + uF2 + vF2.

2010 Mathematics Subject Classifications: 94B05, 94B15, 94B60.

Key Words and Phrases: Gray map, Cyclic codes, Quasi-cyclic code.

1. Introduction

It was introduced linear (1 + u) constacyclic codes and cyclic codes over F2 + uF2 and

characterized codes over F2 which are the Gray images of (1+ u) constacyclic codes or cyclic

codes over F2+uF2, in [6]. In [1], they extended the result of [6] to codes over the commutative

ring Fpk + uFpk where p is a prime, k ∈ N and u2 = 0.

In [5], it was introduced (1−u2)-cyclic codes over F2+uF2+u2F2 and characterized codes

over F2 which are the Gray images of (1−u2)-cyclic codes or cyclic codes over F2+uF2+u2F2.

In [2], it was defined a distance preserving map from F2+uF2+u2F2+u3F2+ . . .+umF2 to

F2 and characterized codes over F2 which are the Gray images of (1−um)-cyclic codes or cyclic

codes over F2+uF2+u2F2+u3F2+ . . .+umF2. In [8], Udomkavanich and Jitman generalized

these results to the ring Fpk +uFpk + . . .+umFpk . The Gray images of (1−um)-constacyclic and

cyclic codes over Fpk + uFpk + . . .+ umF k
p were studied in the mentioned paper.

In [4], (1+ v)-constacyclic codes over R2 = F2+uF2+ vF2+uvF2,u2 = v2 = 0,uv− vu= 0

were studied. (1+ v)-constacyclic codes over R2 of odd length were characterized with help

of cyclic codes over R2.

∗Corresponding author.

Email addresses: abdullah.dertli@gmail.com (A. Dertli), ycengellenmis@yahoo.com (Y. Cengellenmis)

http://www.ejpam.com 305 c© 2016 EJPAM All rights reserved.



A. Dertli, Y. Cengellenmis / Eur. J. Pure Appl. Math, 9 (2016), 305-313 306

In [3], it is studied (1+ u)-cyclic codes over a finite commutative ring

F2+uF2+ vF2+uvF2,u2 = 0, v2 = 0,uv− vu= 0. A set of generator of such constacyclic codes

for an arbitrary length was determined.

In [7], they studied linear codes over a new ring

S = F2 + uF2 + vF2 + uvF2,u2 = 0, v2 = v,uv = vu. It is obtained MacWilliams identities for

Lee weight enumerator of linear codes over this ring using a Gray map from Sn to (F2+uF2)
n.

Moreover, they studied self dual and cyclic codes over S.

Liu Xiusheng and Liu Hualu gave rise to a new ring

R = F2 + uF2 + vF2,u2 = 0, v2 = v,uv = vu = 0 in [9]. It is Frobenius ring. They defined

a Gray map. The MacWilliams identity over F2 and the MacWilliams identities for the Lee

weight enumerators of linear codes over the ring F2 + uF2 + vF2 were given. Moreover, they

gave some examples.

In this paper, it is given some definitions in section 2. It is seen that the image of a (1+u)-

cyclic code of length n over R under the map φ1,1 is a distance invariant cyclic code of length

2n over F2 + vF2. It is shown that if n is odd, then the image of a cyclic code of length n over

R under the map φ1,1 is a permutation equivalent to cyclic code of length 2n over F2+ vF2. In

section 3, it is given a representation of a linear code of length n over R by means of C1 and

C2 which are linear codes of length n over F2+uF2. In section 4, it is characterized codes over

F2 which are the Gray images of (1+ u)-cyclic codes or cyclic codes over F2 + uF2 + vF2. It is

proved that the Gray image of a linear (1+ u)-cyclic code over F2 + uF2 + vF2 of length n is

a binary permutation equivalent to quasi-cyclic codes of index 3 and length 3n over F2. It is

also proved that if n is odd, then every code over F2 which is the Gray image of a linear cyclic

code of length n over F2 + uF2 + vF2 is permutation equivalent to a quasi-cyclic code of index

3.

2. Preliminaries

In [9], the commutative ring R = F2 + uF2 + vF2,u2 = 0, v2 = v,uv = vu = 0 is given.

Then R is a finite, principal ideal and semilocal ring with two maximal ideals Iu+v and I1+v .

The quotient rings R/Iu+v and R/I1+v are isomorphic to F2. A direct decomposition of R is

R= Iv ⊕ I1+v. The set of units of R is R∗ = {1,1+ u}.
Let the C be a code of length n over R and P(C) be its polynomial representation, i.e,

P(C) = {
∑n−1

i=0 ri x
i |(r0, . . . , rn−1) ∈ C} Let σ and ν be maps from Rn to Rn given by

σ(r0, . . . , rn−1) = (rn−1, r0, . . . , rn−2)

and

ν(r0, . . . , rn−1) = ((1+ u)rn−1, r0, . . . , rn−2)

Then C is said to be cyclic if σ(C) = C and (1+ u)− cyclic if ν(C) = C .

A code C of length n over R is cyclic if and only if P(C) is an ideal of R[x]/〈xn−1〉. A code

C of length n over R is (1+ u)− cyclic if and only if P(C) is an ideal of R[x]/〈xn − (1+ u)〉.
Let a ∈ F3n

2 with a = (a0, a1, . . . , a3n−1) = (a
(0)|a(1)|a(2)), a(i) ∈ F n

2 for all i = 0,1,2. Let

σ⊗3 be the map from F3n
2 to F3n

2 given by σ⊗3(a) = (σ̃(a(0))|σ̃(a(1))|σ̃(a(2))) where σ̃ is the



A. Dertli, Y. Cengellenmis / Eur. J. Pure Appl. Math, 9 (2016), 305-313 307

usual cyclic shift

(c0, . . . , cn−1) 7−→ (cn−1, c0, . . . , cn−2)

on F n
2 . A code C̃ of length 3n over F2 is said to be quasi-cyclic of index 3 if σ⊗3(C̃) = C̃ .

The Hamming weight wH(x) of a codeword x is the number of nonzero components in x .

The Hamming distance d(x , y) between two codewords x and y is the Hamming weight of

the codewords x − y . The minimum Hamming distance dH of C is defined as

min{dH(x , y)|x , y ∈ C , x 6= y}.

Let x = (x1, . . . , xn) and y = (y1, . . . , yn) be two vectors of Rn. The Euclidean inner product

of x and y is defined

x y =

n
∑

i=1

x i yi .

The dual code C⊥ of C is defined as C⊥ = {x ∈ Rn|xc = 0 for all c ∈ C}. C is said to be self

orthogonal if C ⊆ C⊥ and C is said to be self dual if C = C⊥.

Recall that the Gray map φ1 on F2 + uF2,u2 = 0 is defined as φ1(z) = (r, r + q) where

z = q+ur with r,q ∈ F2 and the Gray mapφ2 on F2+vF2, v2 = v is defined asφ2(s) = (m, m+t)

where s = m+ vt with m, t ∈ F2. The maps φ1 and φ2 can be extended to (F2 + uF2)
n and

(F2 + vF2)
n, respectively as follows,

φ1 :(F2 + uF2)
n→ F2n

2

(z0, . . . , zn−1) 7→ (r0, . . . , rn−1, r0 ⊕ q0, . . . , rn−1 ⊕ qn−1)

φ2 :(F2 + vF2)
n→ F2n

2

(s0, . . . , sn−1) 7→ (m0, . . . , mn−1, m0 ⊕ t0, . . . , mn−1 ⊕ tn−1)

where zi = ri+uqi , si = mi+vt i and qi , ri , mi , t i ∈ F2 for 0≤ i ≤ n−1 and ⊕ is componentwise

addition in F2.

Each element c ∈ R = F2 + uF2 + vF2 can be expressed c = a + ub where a, b ∈ F2 + vF2.

The map φ1,1 is defined as

φ1,1 :Rn→ (F2 + vF2)
2n

(c0, . . . , cn−1) 7→ (b0, . . . , bn−1, b0 + a0, . . . , bn−1 + an−1)

where ci = ai + ubi with ai , bi ∈ F2 + vF2 for 0≤ i ≤ n− 1.

Each element c ∈ R= F2 + uF2 + vF2 can be also expressed c = a′ + vb′ where

a′, b′ ∈ F2 + uF2. The map φ2,1 is defined as

φ2,1 :Rn→ (F2 + uF2)
2n

(c0, . . . , cn−1) 7→ (a
′
0, . . . , a′n−1, b′0 + a′0, . . . , b′n−1 + a′n−1)

where ci = a′
i
+ vb′

i
with a′

i
, b′

i
∈ F2 + uF2 for 0≤ i ≤ n− 1.

A Gray map φ from R to F m
2 which is the composition of φ1,1 and φ2 or φ2,1 and φ1 can

be obtained.



A. Dertli, Y. Cengellenmis / Eur. J. Pure Appl. Math, 9 (2016), 305-313 308

The Lee weights of 0,1,u, 1 + u ∈ F2 + uF2 are 0,1,2,1 respectively. The Lee weights

of 0,1, v, 1 + v ∈ F2 + vF2 are 0,2,1,1 respectively. These Lee weights can be extended to

(F2 + uF2)
n and (F2 + vF2)

n. It is known that φ1 and φ2 are distance-preserving map from

(F2 + uF2)
n (Lee distance) to F2n

2 (Hamming distance) and (F2 + vF2)
n (Lee distance) to F2n

2

(Hamming distance), respectively. For any element a+vb ∈ R with a, b ∈ F2+uF2, it is defined

Lee weight, denoted by wL as wL(a + vb) = wL(b, b + a). The Lee distance of a linear code

over R, denoted by dL(C) is defined as minimum Lee weight of nonzero codewords of C .

φ1 :(F2 + uF2)
n (Lee distance) → F2n

2 (Hamming distance)

φ2 :(F2 + vF2)
n (Lee distance) → F2n

2 (Hamming distance)

φ1,1 :Rn (Lee distance) → (F2 + vF2)
2n (Lee distance)

φ2,1 :Rn (Lee distance) → (F2 + uF2)
2n (Lee distance)

Now, it will be characterized codes over F2 + vF2 which are the images of (1+ u)-cyclic and

cyclic codes over R.

Proposition 1. Let φ1,1 be defined as above. Let ν be (1+ u)-cyclic shift on Rn and σ the cyclic

shift on (F2 + vF2)
2n. Then φ1,1ν= σφ1,1.

Proof. Let z = (z0, . . . , zn−1) ∈ Rn where ci = qi +uri and qi , ri ∈ F2+ vF2 for 0≤ i ≤ n−1.

From definition, we get,

φ1,1(z) = (r0, . . . , rn−1, r0 + q0, . . . , rn−1 + qn−1)

and

σ(φ1,1(z)) = (rn−1 + qn−1, r0, . . . , rn−1, r0 + q0, . . . , rn−2 + qn−2)

On the other hand,

ν(z) = ((1+ u)zn−1, z0, . . . , zn−2) = (qn−1 + u(qn−1 + rn−1),q0 + ur0, . . . ,qn−2 + urn−2)

and

φ1,1(ν(z)) = (qn−1 + rn−1, r0, . . . ,qn−2 + rn−2).

Theorem 1. A linear code C of length n over R is a (1+ u)-cyclic code iff φ1,1(C) is a cyclic code

of length 2n over F2 + vF2.

Proof. If C is (1+ u)-cyclic code, from Proposition 1 we get φ1,1(ν(C)) = σ(φ1,1(C)). So

φ1,1(C) is a cyclic code of length 2n over F2 + vF2. Conversely, if φ1,1(C) is a cyclic code of

length 2n over F2 + vF2, from Proposition 1, we get φ1,1(ν(C)) = σ(φ1,1(C)) = φ1,1(C). By

using φ1,1 is injection, hence ν(C) = C .

Corollary 1. The image of a (1+u)-cyclic code of length n over R under the map φ1,1 is a distance

invariant cyclic code of length 2n over F2 + vF2.



A. Dertli, Y. Cengellenmis / Eur. J. Pure Appl. Math, 9 (2016), 305-313 309

Note that (1+ u)n = 1+ u if n is odd, (1+ u)n = 1 if n is even. In here, it is studied the

properties of (1+ u) cyclic codes of odd length in this section.

Let µ be the map of R[x]/〈xn−1〉 into R[x]/〈xn−(1+u)〉 defined by µ(c(x)) = c((1+u)x).

If n is odd, then µ is a ring isomorphism. Hence I is an ideal of R[x]/〈xn − 1〉 if and only if

µ(I) is an ideal of R[x]/〈xn − (1+ u)〉. If µ̄′ is the map

µ̄′ : Rn→ Rn

z 7→ (z0, (1+ u)z1, (1+ u)2z2, . . . , (1+ u)n−1zn−1)

where zi = qi + uri and ri ,qi ∈ F2 + vF2 for 0≤ i ≤ n− 1, then it also follows that:

Proposition 2. The set C ⊆ Rn is a linear cyclic code if and only if µ̄′(C) is a linear (1+u)-cyclic

code.

Let τ′ be the following permutation of {0,1,2, . . . , 2n−1}with n odd: τ′ = (1, n+1)(3, n+

3) . . . (n− 2,2n− 2). The Nechaev permutation π′ of (F2 + vF2)
2n is defined by

π′(r0, r1, . . . , r2n−1) = (rτ′(0), rτ′(1), . . . , rτ′(2n−1)).

Proposition 3. Assume n odd, let µ̄′ be the permutation of Rn such that

µ̄′(z0, . . . , zn−1) = (z0, (1+ u)z1, . . . , (1+ u)n−1zn−1).

Then φ1,1µ̄
′ = π′φ1,1.

Corollary 2. If C̃ is the Gray image of a linear cyclic code of length n over R, then C̃ is permutation

equivalent to a cyclic code and length 2n over F2 + vF2.

Proof. From Proposition 2, a code C of length n over R is linear cyclic code if and only

if µ̄′(C) is linear (1 + u)-cyclic. From Theorem 1, this is also so if and only if φ1,1(µ̄
′(C)) is

permutation equivalent to a linear cyclic code over F2 + vF2. From Proposition 3, φ1,1(C) is

permutation equivalent to linear cyclic over F2 + vF2.

3. A Representation of a Code over R

In this section, it will be obtained a representation of a linear code of length n over R by

means of C1 and C2 which are linear codes of length n over F2 + uF2.

Theorem 2. The map φ2,1:Rn→ (F2 + uF2)
2n is a linear isometry.

Proof. For any m, k ∈ Rn and s, t ∈ F2 + uF2, it is verified that

φ2,1(sm+ tk) = sφ2,1(m) + tφ2,1(k),

so φ2,1 is linear. For isometry, we get

dL(φ2,1(m),φ2,1(k)) = wL(φ2,1(m− k)) = wL(m− k) = dL(m, k).



A. Dertli, Y. Cengellenmis / Eur. J. Pure Appl. Math, 9 (2016), 305-313 310

Theorem 3. If C is a linear code of length n over R, then φ2,1(C) is a linear code of length 2n

over F2 + uF2.

Proof. It is seen from linearity of φ2,1.

Let A and B be two codes. The direct product and sum of A and B are defined by, respec-

tively

A⊗ B ={(a, b)|a ∈ A, b ∈ B}

A⊕ B ={a+ b|a ∈ A, b ∈ B}.

Theorem 4. If C be a linear code of length n over R, then C = (1+ v)C1⊕ vC2, φ2,1(C) = C1⊗C2

and |C | = |C1||C2| where C1 = {m ∈ (F2 + uF2)
n|m + vt ∈ C for some t ∈ (F2 + uF2)

n} and

C2 = {m+ t ∈ (F2 + uF2)
n|m+ vt ∈ C for some m ∈ (F2 + uF2)

n}.

Proof. Let c = m + vt ∈ C for some m, t ∈ (F2 + uF2)
n. So m ∈ C1, m + t ∈ C2. Hence

c = (1+ v)m+ v(m+ t) ∈ (1+ v)C1 ⊕ vC2. We have C ⊆ (1+ v)C1 ⊕ vC2. On the other hand,

(1 + v)m + v(m + t) ∈ (1 + v)C1 ⊕ vC2 where m ∈ C1 and t ∈ C2, there exist a, b ∈ C and

r,q ∈ (F2 + uF2)
n such that a = m+ vr and b = m+ t + (1+ v)q. As C is linear over R, from

c = (1+ v)a+ vb ∈ C we have (1+ v)C1 ⊕ vC2 ⊆ C .

Theorem 5. A linear code C = (1+ v)C1⊕ vC2 cyclic over R if and only if C1 and C2 are all cyclic

codes over F2 + uF2.

Proof. Let (r0, . . . , rn−1) ∈ C1 and (s0, . . . , sn−1) ∈ C2. Suppose that ci = (1+ v)ri + vsi for

i = 0, . . . , n−1. Let c = (c0, . . . , cn−1) ∈ C . As C is cyclic, it follows that (cn−1, c0, . . . , cn−2) ∈ C .

Note that (cn−1, c0, . . . , cn−2) = (1+ v)(rn−1, r0, . . . , rn−2) + v(sn−1, s0, . . . , sn−2). So

(rn−1, r0, . . . , rn−2) ∈ C1, (sn−1, s0, . . . , sn−2) ∈ C2, that is C1, C2 are cyclic codes over F2 + uF2.

Conversely, let C1, C2 be cyclic codes over F2 + uF2. Let (cn−1, c0, . . . , cn−2) ∈ C where

ci = (1+ v)ri + vsi for i = 0, . . . , n− 1. Then (r0, . . . , rn−1) ∈ C1 and (s0, . . . , sn−1) ∈ C2. Note

that (cn−1, c0, . . . , cn−2) = (1+v)(rn−1, r0, . . . , rn−2)+v(sn−1, s0, . . . , sn−2) ∈ (1+v)C1⊕vC2 = C .

So C is a cyclic code.

Theorem 6. Let C be a linear code of length n over R. Then φ2,1(C
⊥) = (φ2,1(C))

⊥.

Proof. By using φ2,1(C
⊥) ⊆ (φ2,1(C))

⊥ and |φ2,1(C
⊥)| = |(φ2,1(C))

⊥|, we have expected

result.

Theorem 7. If C is a linear code of length n over R such that C = (1+ v)C1 ⊕ vC2, then

C⊥ = (1+ v)C⊥1 ⊕ vC⊥2 . Moreover C is self dual if and only if C1, C2 are self dual over F2 + uF2.

Theorem 8. Let C = (1+ v)C1 ⊕ vC2 be a linear code of length n over R. Then

dmin(C) = min{d1, d2} where dmin, d1 and d2 are minimum Lee distance of C , C1 and C2, respec-

tively.



A. Dertli, Y. Cengellenmis / Eur. J. Pure Appl. Math, 9 (2016), 305-313 311

4. The Gray Images of (1+ u)− Cyclic Codes and Cyclic over F2 + uF2 + vF2

In this section, by using the Gray map which is defined by Liu Xiusheng, Liu Hualu, we will

characterize codes over F2 which are the Gray images of (1+u)-cyclic and cyclic codes over R.

In [9], it was defined the Gray map φ on Rn as follows

φ :R→ F3
2

a+ ub+ vc 7→ (c, b+ c, a+ b+ c).

This map can be extended to Rn in a natural way. For z = (z0, . . . , zn−1) ∈ Rn,

φ :Rn→ F3n
2

z = (z0, . . . , zn−1) 7→ (s0, . . . , sn−1, s0 ⊕ q0, . . . , sn−1 ⊕ qn−1, r0 ⊕ q0 ⊕ s0, . . . , rn−1 ⊕ qn−1 ⊕ sn−1)

where zi = ri + uqi + vsi , for 0≤ i ≤ n− 1 and ⊕ is componentwise addition in F2.

In [9], they extended the definition of the Lee weight from F2+vF2 to the ring F2+uF2+vF2.

The Lee weight wL(x) of a codeword x = (x1, . . . , xn) was defined as
∑n

i=1 wL(x i) where

wL(x) =













0 if x i = 0

1 if x i = 1,1+ u,u+ v

2 if x i = u, 1+ v, 1+ u+ v

3 if x i = v

The Lee distance dL(x , y) between two codewords x and y is the Lee weight of x− y . The

Gray map φ is an isometry from (Rn, dLee) to F3n
2 under the Hamming distance.

Proposition 4. φν = ρσ⊗3φ where ρ is a permutation of {0, . . . , 3n − 1} which is defined

ρ = (n+ 1,2n+ 1).

Proof. Let z = (z0, z1, . . . , zn−1) ∈ Rn. Let ri ,qi , si ∈ F2 such that zi = ri + uqi + vsi , for

0≤ i ≤ n− 1. We have

φ(z) = (s0, . . . , sn−1, s0 ⊕ q0, . . . , sn−1 ⊕ qn−1, r0 ⊕ q0 ⊕ s0, . . . , rn−1 ⊕ qn−1 ⊕ sn−1).

Then

σ⊗3(φ(z)) =(sn−1, s0, . . . , sn−2, sn−1 ⊕ qn−1, s0 ⊕ q0, . . . , sn−2 ⊕ qn−2

, rn−1 ⊕ qn−1 ⊕ sn−1, r0 ⊕ q0 ⊕ s0, . . . , rn−2 ⊕ qn−2 ⊕ sn−2).

On the other hand, ν(z) = ((1+ u)zn−1, z0, . . . , zn−2) where

(1+ u)zn−1 = rn−1 + u(rn−1 + qn−1) + vsn−1. We have

φ(ν(z)) =(sn−1, s0, . . . , sn−2, rn−1 ⊕ qn−1 ⊕ sn−1,q0 ⊕ s0, . . . ,qn−2 ⊕ sn−2, rn−1 ⊕ qn−1

, r0 ⊕ q0 ⊕ s0, . . . , rn−2 ⊕ qn−2 ⊕ sn−2).

Hence φν= ρσ⊗3φ.

So we have the following theorem.



A. Dertli, Y. Cengellenmis / Eur. J. Pure Appl. Math, 9 (2016), 305-313 312

Theorem 9. A code C of length n over R is (1 + u)−cyclic if and only if φ(C) is permutation

equivalent to quasi-cyclic of index 3 and length 3n over F2.

Proof. Suppose C is (1 + u)−cyclic. As ρ(σ⊗3(φ(C))) = φ(ν(C)), φ(C) is permutation

equivalent to a quasi-cyclic of index 3. Conversely, if φ(C) is permutation equivalent to quasi-

cyclic of index 3, then φ(ν(C)) = ρ(σ⊗3(φ(C))) = φ(C). Since φ is isometry, so ν(C) = C ,

that is C is (1+ u)− cyclic code.

Note that (1+ u)n = 1+ u if n is odd, (1+ u)n = 1 if n is even. In here, it is studied the

properties of (1+ u) cyclic codes of odd length in this section.

Let µ be the map of R[x]/〈xn−1〉 into R[x]/〈xn−(1+u)〉 defined by µ(c(x)) = c((1+u)x).

If n is odd, then µ is a ring isomorphism. Hence I is an ideal of R[x]/〈xn − 1〉 if and only if

µ(I) is an ideal of R[x]/〈xn − (1+ u)〉. If µ̄ is the map

µ̄ :Rn→ Rn

z 7→ (z0, (1+ u)z1, (1+ u)2z2, . . . , (1+ u)n−1zn−1)

where zi = si + ut i + v yi and si, t i , yi ∈ F2 for 0≤ i ≤ n− 1, then it also follows that:

Proposition 5. The set C ⊆ Rn is a linear cyclic code if and only if µ̄(C) is a linear (1+ u)-cyclic

code.

Let τ be the following permutation of {0,1,2, . . . , 3n− 1} with n odd:

τ= (n+ 1,2n+ 1)(n+ 3,2n+ 3)(n+ 5,2n+ 5) . . . (2n− 2,3n− 2)

The permutation π of F3n
2 is defined by

π(r0, r1, . . . , r3n−1) = (rτ(0), rτ(1), . . . , rτ(3n−1))

Proposition 6. Assume n odd, let µ̄ be the permutation of Rn such that

µ̄(z0, . . . , zn−1) = (z0, (1+ u)z1, . . . , (1+ u)n−1zn−1).

Then φµ̄= πφ.

Corollary 3. If C̃ is the Gray image of a linear cyclic code of length n over R, then C̃ is permutation

equivalent to a quasi-cyclic code of index 3 and length 3n over F2.

Proof. From Proposition 5, a code C of length n over R is linear cyclic code if and only

if µ̄(C) is linear (1 + u)-cyclic. From Theorem 9, this is also so if and only if φ(µ̄(C)) is

permutation equivalent to a linear quasi-cyclic code of index 3 over F2. From Proposition 6,

φ(C) is permutation equivalent to linear quasi cyclic of index 3 over F2.



REFERENCES 313

5. Conclusion

It is introduced (1+u)-cyclic codes and cyclic codes over R. Firstly, it is characterized codes

over F2 + vF2 which are the Gray images of (1+ u)-cyclic codes and cyclic codes over R. It is

obtained a representation of a linear code of length n over R by means of C1 and C2 which are

linear codes of length n over F2 + uF2. It is characterized codes over F2 which are the Gray

images of (1+ u)cyclic codes or cyclic codes over R.

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