ArticlePDF Available

A family of constacyclic codes over F 2 + uF 2 + vF 2 + uvF 2

October 2012
Journal of Systems Science and Complexity 25(5)

Authors:

Hefei University of Technology

This paper studies (1 + u)-constacyclic codes over the ring F 2 + uF 2 + vF 2 + uvF 2. It is proved that the image of a (1+u)-constacyclic code of length n over F 2+uF 2+vF 2+uvF 2 under a Gray map is a distance invariant binary quasi-cyclic code of index 2 and length 4n. A set of generators of such constacyclic codes for an arbitrary length is determined. Some optimal binary codes are obtained directly from (1 + u)-constacyclic codes over F 2 + uF 2 + vF 2 + uvF 2. © 2012 Institute of Systems Science, Academy of Mathematics and Systems Science, CAS and Springer-Verlag Berlin Heidelberg.

Content uploaded by Liqi Wang

Content may be subject to copyright.

J Syst Sci Complex (2012) 25: 1032–1040

A FAMILY OF CONSTACYCLIC CODES OVER

F2+uF 2+vF 2+uvF 2∗

Xiaoshan KAI ·Shixin ZHU ·Liqi WANG

DOI: 10.1007/s11424-012-1001-9

Received: 4 January 2011 / Revised: 13 November 2011

The Editorial Oﬃce of JSSC & Springer-Verlag Berlin Heidelberg 2012

Abstract This paper studies (1 + u)-constacyclic codes over the ring F2+uF2+vF2+uvF2.Itis

proved that the image of a (1 + u)-constacyclic code of length nover F2+uF2+vF2+uvF2under a Gray

map is a distance invariant binary quasi-cyclic code of index 2 and length 4n. A set of generators of

such constacyclic codes for an arbitrary length is determined. Some optimal binary codes are obtained

directly from (1 + u)-constacyclic codes over F2+uF2+vF2+uvF2.

Key words Constacyclic code, generator polynomial, Gray distance, Gray map.

1 Introduction

Codes over ﬁnite rings were initiated by Blake in the early 1970s[1−2]. Great progress has

been made in the 1990s because of the signiﬁcant discovery that certain good nonlinear binary

codes can be constructed from cyclic codes over Z4via the Gray map[3]. Meanwhile, one ﬁnds

that codes over ﬁnite rings can be used to design space-time codes and error-correcting coding

schemes for wireless communication systems[4]. Since then, codes over ﬁnite rings have been

studied by many authors[5−8]. In these studies, the ground rings associated with codes are ﬁnite

chain rings in general, and linear codes over this class of ﬁnite rings have been characterized in

several literatures[8−10]. Recently, linear and cyclic codes over the ring F2+uF2+vF2+uvF2

have been considered by Yildiz and Karadenniz in [11] and [12], and some good binary codes

have been obtained as the images under two Gray maps. Because the ring F2+uF2+vF2+uvF2

is not a ﬁnite chain ring, some techniques used in these literatures are diﬀerent from those in

the previous papers. It seems to become more diﬃcult to deal with codes over this ring.

In this paper, we focus on constacyclic codes over the ring F2+uF2+vF2+uvF2,whereu2=

v2=0anduv =vu. We investigate a family of constacyclic codes over F2+uF2+vF2+uvF2,

that is, (1 + u)-constacyclic codes over F2+uF2+vF2+uvF2. Constacyclic codes over ﬁnite

Xiaoshan KAI ·Shixin ZHU

School of Mathematics, Hefei University of Technology, Hefei 230009, China;National Mobile Communications

Research Laboratory, Southeast University, Nanjing 210096, China.

Email: kxs6@sina.com; zhushixin@hfut.edu.cn.

Liqi WANG

School of Mathematics, Hefei University of Technology, Hefei 230009, China.

Email: liqiwangg@163.com.

∗This research was supported by the National Natural Science Foundation of China under Grant No. 60973125,

the Natural Science Foundation of Anhui Province under Grant No. 1208085MA14, and the Fundamental

Research Funds for the Central Universities under Grants Nos. 2012HGXJ0040 and 2011HGBZ1298.

This paper wa s recom me nded f or publ ica tion by E di t or Le i HU .

AFAMILYOFCONSTACYCLICCODESOVERF2+uF2+vF2+uvF21033

commutative rings were ﬁrst introduced by Wolfmann in [13], where it was proved that the

binary image of a linear negacyclic code over Z4is a binary cyclic code (not necessarily linear).

Later, Ling and Blackford extended most of the results in [13–14] to the ring Zpk+1 in [8]. Since

2002, constacyclic codes over various types of ﬁnite chain rings have been extensively studied

(see, for example, [15–21]). In this work, using (1 + u)-constacyclic codes over F2+uF2,we

determine the generator polynomials of (1 + u)-constacyclic codes over F2+uF2+vF2+uvF2

for an arbitrary length. We prove that the image of a (1 + u)-constacyclic code of length n

over F2+uF2+vF2+uvF2under a Gray map is a distance invariant binary quasi-cyclic code

of index 2 and length 4n. Some optimal binary codes are constructed directly as the binary

images of (1 + u)-constacyclic codes over F2+uF2+vF2+uvF2under a Gray map.

2GrayMapsand(1+u)-Constacyclic Codes over F2+uF 2+vF 2+uvF 2

The ring R=F2+uF2+vF2+uvF2is a ﬁnite commutative ring of order 16 with charac-

teristic 2, and the indeterminates u, v meet u2=v2=0anduv =vu[11]. The ring Ris a local

ring, but not a principal ideal ring. Its unique maximal ideal is

I={0,u,v,u+v, uv, u +uv , v +uv, u +v+uv}.

AcodeoverRof length nis a nonempty subset of Rn, and a code is linear over Rof length n

if it is an R-submodule of Rn. For some ﬁxed unit λof R,theλ-constacyclic shift τλon Rnis

the shift τλ(c0,c

1,···,c

n−1)=(λcn−1,c

0,···,c

n−2), and a linear code Cof length nover Ris

λ-constacyclic if the code is invariant under the λ-constacyclic shift τλ. We identify a codeword

c=(c0,c

1,···,c

n−1) with its polynomial representation c(x)=c0+c1x+··· +cn−1xn−1.

Then xc(x) corresponds to a λ-constacyclic shift of c(x) in the ring R[x]/xn−λ.Thus,

λ-constacyclic codes of length nover Rcan be identiﬁed as ideals in the ring R[x]/xn−λ.

Recall that the Gray map φ1on F2+vF2is deﬁned as φ1(z)=(q, q +r)wherez=r+vq

with r, q ∈F2[22] .Themapφ1can be extended to (F2+vF2)nas follows:

φ1:(F2+vF2)n→F2n

(z0,z

1,···,z

n−1)→ (q0,q

1,···,q

n−1,q

0+r0,q

1+r1,···,q

n−1+rn−1),

where zi=ri+vqiwith ri,q

i∈F2for 0 ≤i≤n−1. Every polynomial c(x)∈(F2+vF2)[x]of

degree less than ncanbeexpressedasr(x)+vq(x), where r(x),q(x) are binary polynomials of

degree less than n. The polynomial representation of φ1(c)isφ1(c(x)) = r(x)xn+q(x)(xn+1).

Now, we deﬁne a map φ2from Rnto (F2+vF2)2n, which is a generalization of the Gray map

on (F2+vF2)n. First note that each element c∈Rcan be expressed as c=a+ub,where

a, b ∈F2+vF2.Themapφ2is deﬁned as φ2(c)=(b, b +a). Clearly, this map can be also

extended to Rnas follows:

φ2:Rn→(F2+vF2)2n

(c0,c

1,···,c

n−1)→ (b0,b

1,···,b

n−1,b

0+a0,b

1+a1,···,b

n−1+an−1),

where ci=ai+ubiwith ai,b

i∈F2+vF2for 0 ≤i≤n−1. Hence, for any element

a+ub +vc +uvd ∈Rwith a, b, c, d ∈F2, we can obtain a map φfrom Rto F4

2which is the

composition of φ1and φ2:

φ(a+ub +vc +uvd)=φ1φ2(a+ub +vc +uvd)=(d, c +d, b +d, a +b+c+d).

The Lee weights of 0,1,u(v),1+u(1 + v)∈F2+uF2(F2+vF2)are0,1,2,1, respectively. These

Lee weights can be extended to (F2+uF2)nand (F2+vF2)nin the usual way. It is known that

1034 XIAOSHAN KAI ·SHIXIN ZHU ·LIQI WANG

φ1is a distance-preserving map from (F2+vF2)n(Lee distance) to F2n

2(Hamming distance).

For any element a+vb ∈Rwith a, b ∈F2+uF2, we deﬁne the Gray weight, denoted by wG,

as wG(a+vb)=wL(b, b +a). The Gray distance of a linear code over R, denoted by dG(C),

is deﬁned as the minimum Gray weight of nonzero codewords of C.Thus,wehaveobtained

three distance-preserving linear maps as follows.

φ1:((F2+vF2)n,Lee distance) →(F2n

2,Hamming distance),

φ2:(Rn,Gray distance) →((F2+vF2)2n,Lee distance),

φ:(Rn,Gray distance) →(F4n

2,Hamming distance).

Note that the map φfrom Rnto F4n

2is equivalent to that in [11].

Lemma 2.1 Let φ2be de ﬁned as above. Let τbe the (1 + u)-constacyclic shift on Rnand

σthe cyclic shift on (F2+vF2)2n.Thenφ2τ=σφ2.

Proof Let c=(c0,c

1,···,c

n−1)∈Rn.Letci=ai+ubiwhere ai,b

i∈F2+vF2and

0≤i≤n−1. From deﬁnitions, we have

φ2(c)=(b0,b

1,···,b

n−1,a

0+b0,a

1+b1,···,a

n−1+bn−1),

hence,

σφ2(c)=(an−1+bn−1,b

0,···,b

n−1,a

0+b0,···,a

n−2+bn−2).

On the other hand,

τ(c) = ((1 + u)cn−1,c

0,···,c

n−2)

=(an−1+u(an−1+bn−1),a

0+ub0,···,a

n−2+ubn−2).

Thus

φ2τ(c)=(an−1+bn−1,b

0,···,b

n−2,b

n−1,a

0+b0,···,a

n−2+bn−2).

The result follows.

Theorem 2.2 AlinearcodeCof length nover Ris a (1 + u)-constacyclic code if and

only if φ2(C)isacycliccodeoflength2nover F2+vF2.

Proof If Cis (1 + u)-constacyclic, then using Lemma 2.1 we have

σ(φ2(C)) = φ2(τ(C)) = φ2(C).

Hence, φ2(C) is a cyclic code of length 2nover F2+vF2.Conversely,ifφ2(C) is a cyclic code

of length 2nover F2+vF2, then using Lemma 2.1 again we get

φ2(τ(C)) = σ(φ2(C)) = φ2(C).

Note that φ2is injection, so τ(C)=C.

Thus, we immediately have the following result.

Corollary 2.3 The image of a (1 + u)-constacyclic code of length nover Runder the map

φ2is a distance invariant cyclic code of length 2nover F2+vF2.

Since the indeterminates u, v are symmetrical in the ring R,F2+vF2is similar to F2+uF2.

Corollary 2.3 provides an approach to determine the structure of cyclic codes over F2+uF2by

using (1 + v)-constacyclic codes over R.

AFAMILYOFCONSTACYCLICCODESOVERF2+uF2+vF2+uvF21035

Let σbe the cyclic shift. For any positive integer s,letσsbe the quasi-shift given by

σs(a(1) |a(2) |··· | a(s))=σ(a(1))|σ(a(2) )| ··· |σ(a(s)),

where a(1),a

(2),···,a

(s)∈F2n

2and “ |” denotes the usual vector concatenation. A binary

quasi-cyclic code Cof index sand length 2ns is a subset of (F2n

2)ssuch that σs(C)=C.

Lemma 2.4 Let φbe deﬁ ned as above an d let τbe the (1 + u)-constacyclic shift on Rn.

Then φτ =σ2φ.

Proof Let r=(r0,r

1,···,r

n−1)∈Rn.Letri=ai+ubi+vci+uvdiwhere ai,b

i,c

i,d

i∈F2

and 0 ≤i≤n−1. From deﬁnitions, we have

φ(r)=(d0,d

1,···,d

n−1,c

0+d0,c

1+d1,···,c

n−1+dn−1,

b0+d0,b

1+d1,···,b

n−1+dn−1,a

0+b0+c0+d0,

a1+b1+c1+d1,···,a

n−1+bn−1+cn−1+dn−1),

and so

σ2φ(r)=(cn−1+dn−1,d

0,···,d

n−1,c

0+d0,···,c

n−2+dn−2,

an−1+bn−1+cn−1+dn−1,b

0+d0,···,b

n−1+dn−1,

a0+b0+c0+d0,···,a

n−2+bn−2+cn−2+dn−2).

On the other hand,

τ(r) = ((1 + u)rn−1,r

0,···,r

n−2)

=(an−1+u(an−1+bn−1)+vcn−1+uv(cn−1+dn−1),

a0+ub0+vc0+uvd0,···,a

n−2+ubn−2+vcn−2+uvdn−2).

Hence,

φτ(r)=(cn−1+dn−1,d

0,···,d

n−1,c

0+d0,···,c

n−2+dn−2,

an−1+bn−1+cn−1+dn−1,b

0+d0,···,b

n−1+dn−1,

a0+b0+c0+d0,···,a

n−2+bn−2+cn−2+dn−2).

This completes the proof.

Theorem 2.5 AlinearcodeCof length nover Ris a (1 + u)-constacyclic code if and

only if φ(C)is a binary quasi-cyclic code of index 2and length 4n.

Proof If Cis (1 + u)-constacyclic, then using Lemma 2.4 we have

σ2(φ(C)) = φ(τ(C)) = φ(C).

Hence, φ(C) is a binary quasi-cyclic code of index 2 and length 4n.Conversely,ifφ(C)isa

binary quasi-cyclic code of index 2 and length 4n, then using Lemma 2.4 again we get

φ(τ(C)) = σ2(φ(C)) = φ(C).

Also, φis injection, hence τ(C)=C.

Thus, we immediately have the following result.

Corollary 2.6 The image of a (1 + u)-constacyclic code of length nover Runder the map

φis a distance invariant binary quasi-cyclic code of index 2and length 4n.

1036 XIAOSHAN KAI ·SHIXIN ZHU ·LIQI WANG

3(1+u)-Constacyclic Codes over F2+uF 2+vF 2+uvF 2

3.1 (1+u)-Constacyclic Codes over F2+uF 2

For our purposes, we ﬁrst consider (1 + u)-constacyclic codes over F2+uF2and their binary

images. This will help us to determine the structure of (1 + u)-constacyclic codes over Rand

their Gray weights. It was proved that the binary image φ1(C)ofa(1+u)-constacyclic code C

over F2+uF2of length nis a binary linear cyclic code of length 2nin [19]. Let S=F2+uF2

and Sn=(F2+uF2)[x]/xn−(1 + u).Then(1+u)-constacyclic codes of length nover S

are precisely ideals of Sn.Letn=2

kmwhere mis odd. The structure of (1 + u)-constacyclic

codes of length nover Swas given in [18] (also, see [15]).

Theorem 3.1[18] Let mbe odd and xm−1=f1(x)f2(x)···fr(x)be the unique factorization

of xm−1into a product of monic irreducible pairwise coprime polynomials in F2[x].IfCis a

(1 + u)-constacyclic code over Sof length n=2

km(modd),thenC=fk1

1fk2

2···fkr

r,where

0≤ki≤2k+1. Moreover, |C|=2

2n−r

i=1 kideg(fi).

Lemma 3.2 Let Cbe a (1 + u)-constacyclic code over Sof length n=2

km(modd)with

generator polynomial fk1

1fk2

2···fkr

r,wherefi(x)’s are monic basic irreducible divisors of xm−1

in F2[x]and 0≤ki≤2k+1.ThenC∩uSn=uf l1

1fl2

2···flr

r,whereli=ki−min{2k,k

i}for

each i,1≤i≤r.

Proof The result follows from the fact that u=(f1f2···fr)2kin Sn.

In the following, to avoid confusion, we denote the ideal f(x)in Snby f(x)S

n, while the

ideal f(x)in F2[x]/xn−1by f(x)F2

Theorem 3.3 Let Cbe a (1 + u)-constacyclic code over Sof length n=2

km(modd)

with generator polynomial fk1

1fk2

2···fkr

r,wherefi(x)’s are monic basic irreducible divisors of

xn−1in F2[x]and 0≤ki≤2k+1.Thenφ1(C)=fk1

1fk2

2···fkr

rF2

2n.

Proof Let c(x)∈C.Thenc(x) reduced modulo umust be in the binary cyclic code

g(x)F2

n,whereg(x)=fh1

1fh2

2···fhr

rwith hi=min{2k,k

i}.Thenc(x)=a(x)g(x)+ub(x)for

some polynomials a(x),b(x)∈F2[x] of degree less than n. Therefore,

φ1(c(x)) = b(x)+xn(b(x)+a(x)g(x)) = xna(x)g(x)+(xn+1)b(x).

Since g(x)|xn−1, we have φ1(c(x)) ∈g(x)F2

2n.Thus,φ1(C)⊆g(x)F2

2n.Now,lets(x)∈

φ1(C), then s(x)=t(x)g(x)forsomet(x)∈F2[x]. Let k(x)beinF2[x] such that k(x)g(x)=

xn−1inF2[x]. Then s(x)k(x)=k(x)t(x)g(x)=t(x)(xn+1)=φ1(ut(x)). On the other hand,

note that φ1(C) is cyclic and linear which implies s(x)k(x)∈φ1(C). Hence, ut(x)∈C.By

Lemma 3.2, ut(x)∈C∩uSn=uf l1

1fl2

2···flr

r,whereli=ki−min{2k,k

i}. This implies

t(x)=d(x)fl1

1fl2

2···flr

r,forsomed(x)∈F2[x], so

s(x)=d(x)fl1

1fl2

2···flr

r·fh1

1fh2

2···fhr

r=d(x)fk1

1fk2

2···fkr

This gives φ1(C)⊆fk1

1fk2

2···fkr

rF2

2n. Comparing the cardinality, we have the result.

According to Theorem 3.3, we can determine the Lee distance of a (1 + u)-constacyclic code

over Sof length nusing the Hamming distance of the corresponding binary cyclic code.

3.2 (1+u)-Constacyclic Codes over F2+uF 2+vF 2+uvF 2

Let Rn=R[x]/xn−(1+u).Then(1+u)-constacyclic codes of length nover Rare precisely

ideals of Rn. Consider the homomorphism ϕ:R→Sdeﬁned by ϕ(a+ub +vc+uvd)=a+ub.

The map ϕextends naturally to a ring homomorphism ϕ:Rn→S

ndeﬁned by

ϕ(c0+c1x+···+cn−1xn−1)=ϕ(c0)+ϕ(c1)x+···+ϕ(cn−1)xn−1.

AFAMILYOFCONSTACYCLICCODESOVERF2+uF2+vF2+uvF21037

ForalinearcodeCof length nover R, we can deﬁne two linear codes of length nover Sas

follows.

1) The torsion code Tor(C)={x∈Sn|vx ∈C};

2) The residue code Res(C)={x∈Sn|∃ y∈Sn:x+vy ∈C}.

Acting ϕon C, we get a ring homomorphism

ϕ:C→Res(C),ϕ(a+vb)=a,

where a,b∈Sn.NotethatKerϕ∼

=Tor(C)andϕ(C)=Res(C). By the ﬁrst isomorphism

theorem of ﬁnite groups, we have |C|=|Tor( C)||Res(C)|.

Now, assume that xm−1=f1(x)f2(x)···fr(x) is the unique factorization of xm−1intoa

product of irreducible pairwise coprime polynomials in F2[x]. Let Cbe a (1 + u)-constacyclic

code of length n=2

kmover R, i.e., an ideal of Rn. The image of Cunder the map ϕis an ideal

of Sn. By Theorem 3.1, we have Imϕ=f(x),wheref(x)=fk1

1fk2

2···fkr

rand 0 ≤ki≤2k+1.

On the other hand, Kerϕis also an ideal of Rngenerated by vg(x), where g(x)is an ideal of

Sn. Hence, g(x)=fl1

1fl2

2···flr

r,where0≤li≤2k+1 .So,C=f(x)+vp(x)+uvq(x),vg(x),

where p(x),q(x)∈F2[x]. Clearly, we may assume that deg(g(x)) >deg(p(x)) and deg(g(x)) >

deg(q(x)). Let h(x)=fj1

1fj2

2···fjr

rwith ji=2

k+1 −ki.Notethatf(x)h(x)=0inRn.

Since [f(x)+vp(x)+uvq(x)]h(x)=f(x)h(x)+v[p(x)+uq(x)]h(x)=v[p(x)+uq(x)]h(x)=

v[p(x)+(xn−1)q(x)]h(x)∈Kerϕ=vg(x),wehavethatg(x) divides [p(x)+(xn−1)q(x)]h(x)

in F2[x]. On the other hand, note that v[f(x)+vp(x)+uvq(x)] = vf(x)∈Kerϕ=vg(x),

so g(x) divides f(x). Thus, we have the following result, which describes (1 + u)-constacyclic

codes over Rfor an arbitrary length.

Theorem 3.4 Let f(x)be a divisor of x2n−1in F2[x]and h(x)=(x2n−1)/f(x).IfC

is a (1 + u)-constacyclic code over Rof length n=2

km(modd),thenC=f(x)+vp(x)+

uvq(x),vg(x),whereg(x)|f(x)|x2n−1and g(x)|[p(x)+(xn−1)q(x)]h(x)in F2[x]. Moreover,

deg(g(x)) >deg(p(x)) and deg(g(x)) >deg(q(x)).

Let Cbe a (1 + u)-constacyclic code over Rof length n. ItiseasytoverifythatTor(C)

and Res(C) are both (1 + u)-constacyclic codes of length nover S. The following is obvious.

Corollary 3.5 Let f(x),g(x),p(x)and q(x)be as in Theore m 3.4and C=f(x)+

vp(x)+uvq(x),vg(x)be a (1 + u)-constacyclic code over Rof length n=2

km(modd).Then

Tor (C)=g(x)and Res(C)=f(x).

Theorem 3.6 Le t Cbe the (1 + u)-constacyclic code of length 2km(modd)over Rdeﬁned

in Theorem 3.4,andletd1and d2be the Lee distances of Res(C)and Tor(C), respectively. If

d1≥2d2, then the Gray distance of Cis 2d2.

Proof For any nonzero codeword c∈C, if its entries have elements in Rof the form a+vb

with a∈{1,u,1+u}and b∈S,thenϕ(c)mustbeinRes(C). So, wG(c)≥d1. If its entries

have no elements of the above form, then the element chas the form c=vb with b∈Snand it

must be in vTo r( C). From deﬁnition of the Gray map, the Gray weight of cis equal to 2wL(c).

Hence, if d1≥2d2,thendG(C)=2d2.

Next, we consider a special case. Taking p(x)=q(x) = 0 in Theorem 3.4, we have C=

f(x),vg(x),wheref(x)|x2n−1andg(x)|f(x)inF2[x]. Note that Sis a subring of R,so

Res(C)=f(x)is a subcode of C. On the other hand, vTo r( C)=vg(x)obtained by

multiplying vis contained in C. In this case, the Gray distance of Cis given by min{d1,2d2},

where d1and d2are the Lee distances of the residue and torsion codes, respectively.

Theorem 3.7 Let C=f(x),vg(x)be a (1 + u)-constacyclic code of length 2km(modd)

over R,wheref(x)|x2n−1and g(x)|f(x)in F2[x]. Then the Gray distance of Cis given by

min{d1,2d2},whered1and d2are the Lee distances of Res(C)and Tor(C), respectively.

When nis odd, consider the map μ:

1038 XIAOSHAN KAI ·SHIXIN ZHU ·LIQI WANG

R[x]/xn−1−→R[x]/xn−(1 + u)

deﬁned by μ(a(x)) = a((1 + u)x). Since nis odd, we have

a(x)≡b(x)mod(xn−1)

if and only if

a((1 + u)x)≡b((1 + u)x)mod(xn−(1 + u)).

It is easy to verify that μis a ring isomorphism. Hence, if Ais a subset of R[x]/xn−1and B

is a subset of R[x]/xn−(1 + u)such that μ(A)=B,thenAis an ideal of R[x]/xn−1if and

only if Bis an ideal of R[x]/xn−(1 + u).Equivalently,Ais a cyclic code over Rof length n

if and only if Bis a (1 + u)-constacyclic code over Rof length n. Hence, we can obtain cyclic

codes over Rof odd length nfrom (1 + u)-constacyclic codes over Rof odd length n.

We now consider some (1 +u)-constacyclic codes over Rof diﬀerent lengths and their binary

images.

Table 1 Optimal binary linear codes from (1+ u)-constacyclic

codes over F2+uF2+vF2+uvF2

Length nGenerators f(x),vg(x) Binary image φ

6f(x)=x5+x3+x2+1

vg(x)=vx +v[24,18,4]

6f(x)=ux5+ux4+ux3+ux2+ux +u

vg(x)=vx5+vx3+vx2+v[24,8,8]

6f(x)=ux5+ux4+ux3+ux2+ux +u

vg(x)=vx5+vx4+(v+uv)x3+(v+uv)x2+vx +v[24,4,12]

7f(x)=x5+x4+x3+1

vg(x)=vx +v[28,22,4]

7f(x)=ux4+ux3+ux2+u

vg(x)=vx5+vx4+vx3+v[28,12,8]

7f(x)=ux6+ux5+ux4+ux3+ux2+ux +u

vg(x)=vx6+vx5+vx4+vx3+(v+vu)x2+(v+vu)x+v[28,6,12]

9f(x)=x8+x6+x5+x3+x2+1

vg(x)=vx +v[36,27,4]

10 f(x)=x7+x5+x2+1

vg(x)=vx +v[40,32,4]

12 f(x)=x7+x4+x3+1

vg(x)=vx +v[48,40,4]

14 f(x)=x6+x5+x3+1

vg(x)=vx +v[56,49,4]

15 f(x)=x6+x4+x3+x2+x+1

vg(x)=vx +v[60,53,4]

15 f(x)=0

vg(x)=uvx11 +uvx10 +uvx9+uvx8+uvx6+uvx4+uvx3+uv [60,4,32]

Example 3.8 Consider (1 + u)-constacyclic codes over Rof length 7. In F2[x],

x14 −1=(x−1)2(x3+x+1)

2(x3+x2+1)

AFAMILYOFCONSTACYCLICCODESOVERF2+uF2+vF2+uvF21039

Take f(x)=(x+1)

2(x3+x2+1)(x3+x+1)

2and g(x)=(x+1)

2(x3+x2+ 1), then

C=f(x),vg(x)=ux4+ux3+ux2+u, vx5+vx4+vx3+vis a (1 + u)-constacyclic code

over Rof length 7. By Corollary 3.5, Res(C)=(x+1)

2(x3+x2+1)(x3+x+1)

2is a [7,23,8]

(1 + u)-constacyclic code over F2+uF2,andTor(C)=(x+1)

2(x3+x2+1)is a [7,29,4]

(1 + u)-constacyclic code. Thus, by Theorem 3.7, φ(C)isa[28,12,8] binary quasi-cyclic code,

which is an optimal binary code (see [23]). We list some optimal binary linear codes obtained

from (1 + u)-constacyclic codes over Rin Table 1.

4 Conclusions

We have explored a family of constacyclic codes over the ring F2+uF2+vF2+uvF2, i.e.,

(1 + u)-constacyclic codes over F2+uF2+vF2+uvF2. We have shown that the image of a

(1 + u)-constacyclic code of length nover Runder the map φis a distance invariant quasi-cyclic

code of index 2 and length 4nover F2. We have determined a set of generators of (1 + u)-

constacyclic codes for an arbitrary length. Some optimal binary codes have been obtained as

the binary images of such constacyclic codes over F2+uF2+vF2+uvF2. It would be interesting

to consider other constacyclic codes over F2+uF2+vF2+uvF2. Some more good codes may

be constructed from constacyclic codes over F2+uF2+vF2+uvF2or a more general ring

Fq+uFq+vFq+uvFq(qis a prime power).

References

[1] I.F.Blake,Codesovercertainrings,Inform. Control, 1972, 20: 396–404.

[2] I. F. Blake, Codes over integer residue rings, Inform. Control, 1975, 29: 295–300.

[3] A.R.HammonsJr.,P.V.Kumar,A.R.Calderbank,N.J.A.Sloane,andP.Sol´e, The Z4-

linearity of Kerdock, Preparata, Goethals, and related codes, IEEE Trans. Inform. Theory, 1994,

40: 301–319.

[4] V. Tarokh, N. Seshadri, and A. R. Calderbank, Space-time codes for high data rate wireless com-

munication: Performance criterion and construction, IEEE Trans. Inform. Theory, 1998, 44:

744–765.

[5] A. R. Calderbank and N. J. A. Sloane, Modular and p-adic cyclic codes, Des. Codes Cryptogr.,

1995, 6: 21–35.

[6] H.Q.DinhandS.R.L´opez-Permouth, Cyclic and negacyclic codes over ﬁnite chain rings, IEEE

Trans. Inform. Theory, 2004, 50: 1728–1744.

[7] P. Kanwar and S. R. L´opez-Permouth, Cyclic codes over the integers modulo pm,Finite Fields

Appl., 1997, 3: 334–352.

[8] S. Ling and J. Blackford, Zpk+1 -linear codes, IEEE Trans. Inform. Theory, 2002, 48: 2592–2605.

[9] G. H. Norton and A. Sˇalˇagean, On the structure of linear and cyclic codes over a ﬁnite chain ring,

Appl. Algebra Eng. Comm. Comput., 2000, 10: 489–506.

[10] G. H. Norton and A. Sˇalˇagean, On the Hamming distance of linear codes over a ﬁnite chain ring,

IEEE Trans. Inform. Theory, 2000, 46: 1060–1067.

[11] B. Yildiz and S. Karadenniz, Linear codes over F2+uF2+vF2+uvF2,Des. Codes Cryptogr.,

2010, 54: 61–81.

[12] B. Yildiz and S. Karadenniz, Cyclic codes over F2+uF2+vF2+uvF2,Des. Codes Cryptogr., 2011,

58: 221–234.

[13] J. Wolfmann, Negacyclic and cyclic codes over Z4,IEEE Trans. Inform. Theory, 1999, 45: 2527–

2532.

[14] J. Wolfmann, Binary images of cyclic codes over Z4,IEEE Trans. Inform. Theory, 2001, 47:

1773–1779.

[15] T. Abualrub and I. Siap, Constacyclic codes over F2+uF2,J. Franklin Inst., 2009, 346: 520–529.

1040 XIAOSHAN KAI ·SHIXIN ZHU ·LIQI WANG

[16] T. Blackford, Negacyclic codes over Z4of even length, IEEE Trans. Inform. Theory, 2003, 49:

1417–1424.

[17] H. Q. Dinh, Constacyclic codes of length 2sover Galois extension rings of F2+uF2,IEEE Trans.

Inform. Theory, 2009, 55: 1730–1740.

[18] X. Kai, S. Zhu, and P. Li, (1 + λu)-Constacyclic codes over Fp[u]/um,J. Franklin Inst., 2010,

347: 751–762.

[19] J. Qian, L. Zhang, and S. Zhu, (1 + u)-Constacyclic and cyclic codes over F2+uF2,Appl. Math.

Lett., 2006, 19: 820–823.

[20] A. Sˇalˇagean, Repeated-root cyclic and negacyclic codes over ﬁnite chain rings, Discrete Appl.

Math., 2006, 154: 413–419.

[21] S. Zhu and X. Kai, The Hamming distances of negacyclic codes of length 2sover GR(2a,m),

Journal of Systems Science &Complexity, 2008, 21(1): 60–66.

[22] A. Bonnecaze and P. Udaya, Cyclic codes and self-dual codes over F2+uF2,IEEE Trans. Inform.

Theory, 1999, 45: 1250–1255.

[23] M. Grassl, Bound on the minimum distance of linear codes, http://www.codetables.de.

Some results of linear codes over the ring $\mathbb{Z}_4+u\mathbb{Z}_4+v\mathbb{Z}_4+uv\mathbb{Z}_4

Preprint

Jan 2016

In this paper, we mainly study the theory of linear codes over the ring

R =\mathbb{Z}_4+u\mathbb{Z}_4+v\mathbb{Z}_4+uv\mathbb{Z}_4

. By the Chinese Remainder Theorem, we have R is isomorphic to the direct sum of four rings

\mathbb{Z}_4

. We define a Gray map

\Phi

from

R^{n}

\mathbb{Z}_4^{4n}

, which is a distance preserving map. The Gray image of a cyclic code over

R^{n}

is a linear code over

\mathbb{Z}_4

. Furthermore, we study the MacWilliams identities of linear codes over R and give the the generator polynomials of cyclic codes over R. Finally, we discuss some properties of MDS codes over R.

On a class of constacyclic codes and MDS code construction over 𝔽pm[u,v] 〈u2−δu,v2−γv,uv−vu〉

Article

Sep 2024

Let [Formula: see text] denote a finite field, where [Formula: see text] is a prime and [Formula: see text] is a positive integer. In this paper, we study idempotent decomposition of the finite, commutative, non-local ring, [Formula: see text], where [Formula: see text]. This paper consists of the structure of [Formula: see text]-constacyclic (CC) codes over [Formula: see text], where [Formula: see text]. The structure of dual of CC codes is used to determine the Euclidean hull of such codes. Then, the Hamming distance and symbol-pair distance for the [Formula: see text]-CC codes of length [Formula: see text] over [Formula: see text] is determined. Additionally, conditions for repeated root [Formula: see text]-CC code to be maximal distance separable (MDS) in terms of Hamming distance and symbol-pair distance are provided. Finally, we present examples of several linear and symbol-pair codes some of which are MDS with respect to Hamming and symbol-pair distance.

Constacyclic and Skew Constacyclic Codes Over a Finite Commutative Non-chain Ring

Chapter

Sep 2023

For an odd prime p, this article studies the

\lambda

-constacyclic and skew

\lambda

-constacyclic codes of arbitrary length over the finite commutative non-chain ring

R=\mathbb {F}_{p^m}[u,v,w]/\langle u^{2}-1,v^{2}-1,w^{2}-1,uv-vu,vw-wv,wu-uw\rangle

, where

\lambda

is a unit in R. By using the decomposition method, we determine the structure of

\lambda

-constacyclic and skew

\lambda

-constacyclic codes. Also, the necessary and sufficient conditions of these codes to be self-dual are obtained. Further, it is shown that the Gray images of

\lambda

-constacyclic and skew

\lambda

-constacyclic codes of length n over R are quasi-twisted and skew quasi-twisted codes, respectively of length 8n and index 8 over

\mathbb {F}_{p^m}

. Finally, two non-trivial examples are given to validate the obtained results.

Cyclic codes over rings of matrices

Article

Sep 2022

In this paper, we consider the ring of matrices \begin{document}

\mathcal{A}

\end{document} of order \begin{document} 2 \end{document} over the ring \begin{document}

\mathbb{F}_2 [u] / \langle u^k \rangle

\end{document}, where \begin{document} u \end{document} is an indeterminate with \begin{document}

u^k = 0

\end{document}, i.e. \begin{document}

\mathcal{A} = M_2 ( \mathbb{F}_2 [u] / \langle u^k \rangle)

\end{document}. We derive the structure theorem for cyclic codes of odd length \begin{document} n \end{document} over the ring \begin{document}

\mathcal{A}

\end{document} with the help of isometry map from \begin{document}

\mathcal{A}

\end{document} to \begin{document}

\mathbb{F}_4 [u, v] / \langle u^k, v^2, u v - v u \rangle

\end{document}, where \begin{document} v \end{document} is an indeterminate satisfying \begin{document}

v^2 = 0

\end{document} and \begin{document}

u v = v u

\end{document}. We define a map \begin{document}

\theta

\end{document} which takes the linear codes of odd length \begin{document} n \end{document} over \begin{document}

\mathcal{A}

\end{document} to linear codes of even length \begin{document}

2 k n

\end{document} over \begin{document}

\mathbb{F}_4

\end{document}. We also define a weight on the ring \begin{document}

\mathcal{A}

\end{document} which is an extension of the weight defined over the ring \begin{document}

M_2 ( \mathbb{F}_2)

\end{document}. An example is also given as applications to construct the linear codes of odd length \begin{document} n \end{document} over \begin{document}

\mathcal{A}

\end{document}.

Construction of optimal codes from a class of constacyclic codes

Article

Jan 2022

In this paper, the structure of

(\alpha u+\beta (u-\delta ))

-constacyclic codes of length n over the finite commutative non-local ring

\frac{{\mathbb {F}}_{p^m}[u]}{\left\langle u^2-\delta u \right\rangle }

is provided, where

\alpha , \beta

and

\delta

are units of

{\mathbb {F}}_{p^m}

. The structure of dual constacyclic codes is considered and the hull of all such codes is determined. Using that, the Hamming and symbol-pair distances of

(\alpha u+\beta (u-\delta ))

-constacyclic code over the ring

\frac{{\mathbb {F}}_{p^m}[u]}{\left\langle u^2-\delta u \right\rangle }

are established for code length

p^s

. As applications, the MDS and MDS symbol-pair codes among them are completely identified.

New quantum codes from skew constacyclic codes

Article

Aug 2021
ADV MATH COMMUN

For an odd prime \begin{document} p \end{document} and positive integers \begin{document} m \end{document} and \begin{document}

\ell

\end{document}, let \begin{document}

\mathbb{F}_{p^m}

\end{document} be the finite field with \begin{document}

p^{m}

\end{document} elements and \begin{document}

R_{\ell,m} = \mathbb{F}_{p^m}[v_1,v_2,\dots,v_{\ell}]/\langle v^{2}_{i}-1, v_{i}v_{j}-v_{j}v_{i}\rangle_{1\leq i, j\leq \ell}

\end{document}. Thus \begin{document}

R_{\ell,m}

\end{document} is a finite commutative non-chain ring of order \begin{document}

p^{2^{\ell} m}

\end{document} with characteristic \begin{document} p \end{document}. In this paper, we aim to construct quantum codes from skew constacyclic codes over \begin{document}

R_{\ell,m}

\end{document}. First, we discuss the structures of skew constacyclic codes and determine their Euclidean dual codes. Then a relation between these codes and their Euclidean duals has been obtained. Finally, with the help of a duality-preserving Gray map and the CSS construction, many MDS and better non-binary quantum codes are obtained as compared to the best-known quantum codes available in the literature.

Duadic codes over Fp + uFp + vFp + uvFp

Article

Jan 2023

Duadic codes constitute a well-known class of cyclic codes. In this paper, we study the structure of duadic codes of length n over the ring R = Fp + uFp + vFp + uvFp, u2 = v2 = 0, uv = vu, where p is prime and (n, p) = 1. These codes have been studied here in the setting of abelian codes over R, and we have used Fourier transform and idempotents to study them. We have characterized abelian codes over R by studying their torsion and residue codes. It is shown that the Gray image of an abelian code of length n over R is a binary abelian code of length 4n. Conditions for self-duality and self-orthogonality of duadic codes over R are derived. Some conditions on the existence of self-dual augmented and extended codes over R are presented. We have also studied Type II self-dual augmented and extended codes over R. Some results related to theminimum Lee distances of duadic codes over R are presented. We have also presented a sufficient condition for abelian codes of the same length over R to have the same minimum Hamming distance. Some optimal binary linear codes of length 36 and ternary linear codes of length 16 have been obtained as Gray images of duadic codes of length 9 and 4, respectively, over R using the computational algebra system Magma.

Duality of constacyclic codes of prime power length over the finite non-commutative chain ring F p m [ u , θ ] 〈 u 2 〉

Article

Jun 2022
DISCRETE MATH

Let R=Fpm[u,θ]〈u2〉 where θ is an automorphism of Fpm but it is not the identity. The list of (left and right) α-constacyclic codes of length ps over R is provided where θ(α)=α. The self-dual α-constacyclic codes are given. In addition, we derive the condition of LCD codes for those codes. For the remaining result, the condition of self-orthogonal left α-constacyclic codes is also obtained.

Concise Encyclopedia of Coding Theory

Book

Mar 2021

Negacyclic codes of prime power length over the finite non-commutative chain ring 𝔽pm[u,𝜃] 〈u2〉

Article

Feb 2021

In this paper, we focus on the algebraic structure of left negacyclic codes of length [Formula: see text] over the finite non-commutative chain ring [Formula: see text] where [Formula: see text] is an automorphism on [Formula: see text]. After that, the number of codewords of all left negacyclic codes is obtained. For each left negacyclic code, we also obtain the structure of its right dual code. In the remaining result, the number of distinct left negacyclic codes is given. Finally, a one-to-one correspondence between left negacyclic and left [Formula: see text]-constacyclic codes of length [Formula: see text] over [Formula: see text] is constructed via ring isomorphism, which carries over the results regarding left negacyclic codes corresponding to left [Formula: see text]-constacyclic codes of length [Formula: see text] over [Formula: see text] where [Formula: see text] is a nonzero element of the field [Formula: see text] such that [Formula: see text].

Regular Article: Cyclic Codes over the Integers Modulopm

Article

Full-text available

Oct 1997
FINITE FIELDS TH APP

The purpose of this paper is twofold. First, we generalize the results of Pless and Qian and those of Pless, Sole, and Qian for cyclic Z"4-codes to cyclic Z"p"^"m-codes. Second, we establish connections between this new development and the results on cyclic Z"p"^"m-codes obtained by Calderbank and Sloane. We produce generators for the cyclic Z"p"^"m-codes which are analogs to those for cyclic Z"4-codes. We show that these may be used to produce a single generator for such codes. In particular, this proves that the ringR"n= Z"p"^"m[x]/(x^n- 1) is principal, a result that had been previously announced with an incorrect proof. Generators for dual codes of cyclic Z"p"^"m-codes are produced from the generators of the corresponding cyclic Z"p"^"m-codes. In addition, we also obtain generators for the cyclicp^m-ary codes induced from the idempotent generators for cyclicp-ary codes.

Cyclic and negacyclic codes over finite commutative chain rings

Presentation

Dec 2003

Hai Q. Dinh

Linear codes over 𝔽 2 +u𝔽 2 +v𝔽 2 +uv𝔽 2

Article

Jan 2010

Constacyclic codes over F 2 +uF 2

Article

Jun 2009
J FRANKLIN I

We study the structure of (1+u)-constacyclic codes of an arbitrary length n over the ring F 2 +uF 2 . We find a set of generators for each (1+u)-constacyclic code and its dual. We study the rank of cyclic codes and find their minimal spanning sets. We prove that the Gray image of a (1+u)-constacyclic code is a binary cyclic code of length 2n. We conclude by giving examples of constacyclic codes and their Gray image binary codes. We give a direct construction of a [12,7,4] linear binary cyclic code that match the Hamming distance of the best binary code with length 12 and dimension 7.

Cyclic codes over 𝔽 2 +u𝔽 2 +v𝔽 2 +uv𝔽 2

Article

Jan 2011

Cyclic codes over {{\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2}

Article

Mar 2011

In this work, we focus on cyclic codes over the ring

{{{\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2}}

, which is not a finite chain ring. We use ideas from group rings and works of AbuAlrub et.al. in (Des Codes Crypt 42:273–287, 2007) to characterize the ring

{({{\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2})/(x^n-1)}

and cyclic codes of odd length. Some good binary codes are obtained as the images of cyclic codes over

{{{\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2}}

under two Gray maps that are defined. We also characterize the binary images of cyclic codes over

{{{\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2}}

in general.

(1+u)(1+u) constacyclic and cyclic codes over F2+uF2F2+uF2

Article

Aug 2006
APPL MATH LETT

We introduce (1+u)(1+u) constacyclic and cyclic codes over the ring F2+uF2={0,1,u,ū=u+1}, where u2=0u2=0, and study them by analogy with the Z4Z4 case. We prove that the Gray image of a linear (1+u)(1+u) constacyclic code over F2+uF2F2+uF2 of length nn is a binary distance invariant linear cyclic code. We also prove that, if nn is odd, then every binary code which is the Gray image of a linear cyclic code over F2+uF2F2+uF2 of length nn is equivalent to a linear cyclic code.

Self-dual codes over F2+uF2+vF2+uvF2F2+uF2+vF2+uvF2

Article

Dec 2010
J FRANKLIN I

The focus in this work is on self-dual codes over the ring F2+uF2+vF2+uvF2F2+uF2+vF2+uvF2. Type I and Type II codes over F2+uF2+vF2+uvF2F2+uF2+vF2+uvF2 are defined and some of the techniques used in the literature are applied to get some theoretical results about self-dual codes on F2+uF2+vF2+uvF2F2+uF2+vF2+uvF2 and their binary images. In particular some extremal and optimal binary codes of Type I and Type II including the extended Golay code are obtained as the Gray images of codes over F2+uF2+vF2+uvF2F2+uF2+vF2+uvF2.

(1+λu)-Constacyclic codes over Fp[u]/〈um〉

Article

Jun 2010
J FRANKLIN I

Motivated by the work in [1] of Abualrub and Siap (2009), we investigate (1+λu)(1+λu)-constacyclic codes over Fp[u]/〈um〉Fp[u]/〈um〉 of an arbitrary length, where λλ is a nonzero element of Fp. We find the generator polynomials of (1+λu)(1+λu)-constacyclic codes over Fp[u]/〈um〉Fp[u]/〈um〉, and determine the number of (1+λu)(1+λu)-constacyclic codes over Fp[u]/〈um〉Fp[u]/〈um〉 for a given length, as well as the number of codewords in each such code. Some optimal linear codes over F3 and F5 are constructed from (1+λu)(1+λu)-constacyclic codes over Fp+uFp under a Gray map.

Constacyclic and Cyclic Codes over F2 + uF2 + u2F2

Article

Jun 2006

A new Gray map between codes over F2 + uF2 + u2F2 and codes over F2 is defined. We prove that the Gray image of a linear (1-u2)-cyclic code over F2 + uF2 + u2F2 of length n is a binary distance invariant linear quasi-cyclic code. We also prove that, if n is odd, then every binary code which is the Gray image of a linear cyclic code over F2 + uF2 + u2F2 of length n is equivalent to a quasi-cyclic code.

A family of constacyclic codes over F 2 + uF 2 + vF 2 + uvF 2

Abstract

Recommended publications

Cyclic codes and the weight enumerator of linear codes over F2 + vF2 + v2F2

(1 − uv)-constacyclic codes over $\mathbb{F}_p + u\mathbb{F}_p + v\mathbb{F}_p + uv\mathbb{F}_p

Linear, Cyclic and constacyclic codes over S4 = F2 + uF2 + u2F2 + u3F2

(1+λu)-Constacyclic codes over Fp[u]/〈um〉

A class of constacyclic codes over Fp+vFpFp+vFp and its Gray image