# Quantum Codes Obtained from Skewϱ-λ-Constacyclic Codes over Rk

Authors: Bo Kong; Xiying Zheng; Jie Liu; Hongjing Jiang
Journal: Quantum Engineering
 (2023)

---

## Abstract

LetRk=Fqu1,u2,⋯,uk/<ui2=αiui,uiuj=ujui=0>, where q=pm, p is an odd prime, αi is a unit over Fq, and i,j=1,2,⋯,k. In this article, we define a Gray map from Rkn to Fqk+1n, we study the structure of skew ϱ-λ-constacyclic codes over Rk, and then we give the necessary and sufficient conditions for skew ϱ-λ-constacyclic codes over Rk to satisfy dual containing. Further, we have obtained some new nonbinary quantum codes from skew ϱ-λ-constacyclic over Rk by using the CSS construction.


---

## Body

## 1. Introduction

Since Calderbank and Shor [1] and Steane [2] introduced a simple construction of quantum error-correcting code in 1996, many quantum error-correcting codes have been obtained from classical error-correcting codes by using the CSS construction [3–6]. In recent years, many researchers constructed quantum codes from constacyclic codes over finite nonchain rings [7–14]. In [15, 16], Boucher et al. proposed skew cyclic codes as a new kind of generalized cyclic codes by applying skew polynomial rings. Siap et al. [17] studied the structure of skew cyclic codes for an arbitrary length over finite fields. In [18, 19], skew constacyclic codes were studied over finite fields and finite chain rings. Bag et al. constructed quantum codes from skew 1−2u1−2u2−⋯−2um-constacyclic codes over Fq+u1Fq+⋯+u2mFq and Θ-λ-skew constacyclic codes over Fqu,v/u2−1,v2−1,uv=vu by applying the CSS construction [20, 21]. In [22, 23], some good quantum codes were obtained from linear skew constacyclic over Fq2R and Fq2v1,v2,…,vl/vi2−1,vivj−vjvi1≤i,j≤l by using the Hermitian construction. In [24, 25], some new quantum codes were obtained from skew constacyclic codes over Re,qu/ue=1, and some MDS quantum codes were construed from skew cyclic codes over Fqu/uk+1−u by applying the CSS construction. Dinh et al. [26] obtained some optimal codes and near-optimal codes from skew θ-cyclic codes and discussed the advantages of quantum codes from skew θ-cyclic codes than from cyclic codes over Fq. In this article, we study the algebraic structures of skew ϱ-λ-constacyclic codes over Rk=Fqu1,u2,⋯,uk/ui2=αiui,uiuj=ujui=0; as an application, we give some new quantum codes from skew ϱ-λ-constacyclic codes over Rk by using the CSS construction.The rest of this article is arranged as follows: In Section2, we define a new nonchain ring Rk and a Gray map from Rkn to Fqk+1n and introduce some basic knowledge of skew constacyclic code over Rk. In Section 3, we give the necessary and sufficient conditions for skew ϱ-λ-constacyclic codes over Rk to satisfy dual containing. In Section 4, we give some examples and obtain some new quantum codes from skew ϱ-λ-constacyclic codes over Rk.


## 2. Preliminaries

LetRk=Fqu1,u2,⋯,uk/ui2=αiui,uiuj=ujui=0 be a nonchain ring, where q=pm,p is an odd prime and αi is a unit over Fq,i,j=1,2,⋯,k.Clearly,Rk is semilocal and has qk+1 elements.Letς1=u1/α1,ς2=u2/α2,⋯,ςk=uk/αk,ςk+1=1−u1/α1−u2/α2−⋯−uk/αk. We can get that ςiςj=0, when i≠j, ςiςi=ςi, when i,j=1,2,⋯,k+1, and 1=ς1+ς2+⋯+ςk+1. Thus, Rk=ς1Rk⊕ς2Rk⊕⋯⊕ςk+1Rk.For anyr∈Rk, r can only be said to r=r1ς1+r2ς2+⋯+rk+1ςk+1, where rj∈Fq and j=1,2,⋯,k+1.Letθt be Fq automorphism, θt:Fq⟶Fq by θta=apt. We define the automorphism of Rk as follows:(1)ϱ:Rk→Rk,a0+a1u1+⋯+akuk↦θta0+θta1u1+⋯+θtakuk.By the above definition, the order ofϱ is m/t.Let the setRkx,ϱ=a0+a1x+⋯+anxn,ai∈Rk,i=0,1,2,⋯,n, the addition on Rkx,ϱ is defined as the general form of polynomials and the multiplication of polynomials is axibxj=aϱibxi+j.By the above definition, it is easy to know that the setRkx,ϱ is a noncommutative ring and a skew polynomial ring. ∀fx,gx∈Rkx,ϱ,gx is a right divisor of fx if there exists qx∈Rkx,ϱ subject to fx=qx∗gx. Similarly, the left divisor can be given as above.Letλ be a unit of Rk, the skew constacyclic shift σϱ,λ of c=c0,c1,⋯,cn−1∈Rkn is defined by σϱ,λc=λϱcn−1,ϱc0,⋯,ϱcn−2. Then, C is called a skew ϱ-λ-constacyclic code of length n over Rk if C is invariant under σϱ,λ:Rkn⟶Rkn. In particular, C is called a skew ϱ-cyclic code and skew ϱ-negacyclic code of length n over Rk, when λ=1 and λ=−1.A map is defined as follows:(2)ψ:Rkn→Rkx,ϱxn−λ,a0,a1,⋯,an−1↦a0+a1x+⋯+an−1xn−1.Then,∀a0,a1,⋯,an−1∈Rkn is identified as a polynomial a0+a1x+⋯+an−1xn−1 over Rkx,ϱ/xn−λ. Let the order of ϱ,ϱ=l; if ln, we define a skew ϱ-λ-constacyclic code of length n over Rk as a left ideal of Rkx,ϱ/xn−λ.By the same method of Lemma 7 in [17], we can have the following lemma.Lemma 1.
Iffxgx∈ZRkx,ϱ, the centre of Rkx,ϱ is ZRkx,ϱ and then fxgx=gxfx∈Rkx,ϱ.For anya=a1ς1+a2ς2+⋯+ak+1ςk+1∈Rk, the Gray map ϕk is defined as follows:(3)ϕk:Rk→Fqk+1,a↦a1,a2,⋯,ak+1.We extendϕk as follows:(4)ϕk:Rkn→Fqk+1n,a0,a1,⋯,an−1↦a1,0,⋯,a1,n−1,a2,0,⋯,a2,n−1,⋯,ak+1,0,⋯,ak+1,n−1,where ai=a1,iς1+a2,iς2+⋯+ak+1,iςk+1∈Rk,i=0,1,2,⋯,n−1.For anyc=c0,c1,⋯,cn−1∈C, c can be said to be as follows:(5)c=c0,c1,⋯,cn−1↔cx=∑i=0n−1cixi∈Rkx.LetC be a linear code of length n over Rk, and(6)Cj=xj∈Fqn∑j=1k+1xjςj∈C,x1,x2,⋯,xj−1,xj+1,⋯,xk+1∈Fqn,for j=1,2,⋯,k+1. One can quickly verify that Cj is a linear code of length n over Fq for j=1,2,⋯,k+1, and C=⊕j=1k+1ςjCj,C=∏j=1k+1Cj.Lemma 2 (see [14]).
An elementλ=λ1ς1+λ2ς2+⋯+λk+1ςk+1 is a unit in Rk if and only if λj is a unit in Fq for j=1,2,⋯,k+1.


## 3. Skewϱ-λ-Constacyclic Codes over Rk

Lemma 3.
LetC=⊕j=1k+1ςjCj be a linear code of length n over Rk, and λ=λ1ς1+λ2ς2+⋯+λk+1ςk+1 is a unit in Rk,ordϱ∣n. Then, ϱλ=λ if and only if θtλj=λj and j=1,2,⋯,k+1, where θtαi=αi and i=1,2,⋯,k.Proof.
Supposeϱλ=λ, we have(7)λ=λ1ς1+λ2ς2+⋯+λk+1ςk+1=λk+1+u1λ1−λk+1α1+⋯+ukλk−λk+1αk=ϱλk+1+u1λ1−λk+1α1+⋯+ukλk−λk+1αk=θtλk+1+u1θtλ1−λk+1α1+⋯+ukθtλk−λk+1αk=λk+1pt+u1λ1−λk+1α1pt+⋯+ukλk−λk+1αkpt=λk+1pt+u1λ1pt−λk+1ptα1pt+⋯+ukλkpt−λk+1ptαkpt=θtλk+1+θtλ1−θtλk+1θtα1+⋯+θtλk−θtλk+1θtαk.
On comparing the coefficients, we have(8)λk+1pt=λk+1,θtλ1−θtλk+1θtα1=λ1−λk+1α1,⋯,θtλk−θtλk+1θtαk=λk−λk+1αk.
Note thatθtαi=αi for i=1,2,⋯,k, we can get that θtλj=λj for j=1,2,⋯,k+1.
Conversely, ifθtλj=λj for j=1,2,⋯,k+1, note that θtαi=αi for i=1,2,⋯,k, then we can have θtςi=θtui/αi=ςi and θtςk+1=θt1−u1/α1−⋯−uk/αk=ςk+1.
So,ϱλ=ϱλ1ς1+λ2ς2+⋯+λk+1ςk+1=λ1ς1+λ2ς2+⋯+λk+1ςk+1=λ.Theorem 1.
LetC=⊕j=1k+1ςjCj be a linear code of length n over Rk and λ=λ1ς1+λ2ς2+⋯+λk+1ςk+1 is a unit in Rk,ordϱn,ϱλ=λ. Then, C is a skew ϱ-λ-constacyclic code of length n over Rk if and only if Cj is a skew θt-λj-constacyclic code of length n over Fq for j=1,2,⋯,k+1.Proof.
For anycj=cj,0,cj,1,⋯,cj,n−1∈Cj, j=1,2,⋯,k+1. Then, c=ς1c1+ς2c2+⋯+ςk+1ck+1=∑j=1k+1ςjcj,0,∑j=1k+1ςjcj,1,⋯,∑j=1k+1ςjcj,n−1∈C
IfCj is a θt-λj-constacyclic code of length n over Fq, then(9)σθt,λjcj=σθt,λjcj,0,cj,1,⋯,cj,n−1=λjcj,n−1θt,cj,0θt,⋯,cj,n−2θt∈Cj,σϱ,λc=λ1ς1+λ2ς2+⋯+λk+1ςk+1ϱ∑i=1k+1ςici,n−1,ϱ∑i=1k+1ςici,0,⋯,ϱ∑i=1k+1ςici,n−2=λ1ς1+λ2ς2+⋯+λk+1ςk+1θt∑i=1k+1ςici,n−1,θt∑i=1k+1ςici,0,⋯,θt∑i=1k+1ςici,n−2=λ1ς1+λ2ς2+⋯+λk+1ςk+1∑i=1k+1ςiθtci,n−1,∑i=1k+1ςiθtci,0,⋯,∑i=1k+1ςiθtci,n−2=ς1σθt,λ1c1+ς2σθt,λ2c2+⋯+ςk+1σθt,λk+1ck+1∈C.
So,C is a skew ϱ-λ-constacyclic code of length n over Rk.
On the other hand, ifC is a skew ϱ-λ-constacyclic code of length n over Rk, we have(10)σϱ,λc=ς1σθt,λ1c1+ς2σθt,λ2c2+⋯+ςk+1σθt,λk+1ck+1∈C=⊕j=1k+1ςjCj.
So,σθt,λjcj∈Cj,Cj is a skew θt-λj-constacyclic code of length n over Fq for j=1,2,⋯,k+1.Theorem 2.
LetC=⊕j=1k+1ςjCj be a skew ϱ-λ-constacyclic code of length n over Rk,λ=λ1ς1+λ2ς2+⋯+λk+1ςk+1 is a unit in Rk,ordϱn,ϱλ=λ. Then, C⊥=∑j=1k+1ςjCj⊥ is a skew θt-λ−1-constacyclic code of length n over Rk, and Cj⊥ is a skew θt-λj−1-constacyclic code over Fq for j=1,2,⋯,k+1, where λ−1=λ1−1ς1+λ2−1ς2+⋯+λk+1−1ςk+1.Proof.
LetC=⊕j=1k+1ςjCj be a skew ϱ-λ-constacyclic code of length n over Rk, where λ=λ1ς1+λ2ς2+⋯+λk+1ςk+1 is a unit in Rk. Foranyx=x0,x1,⋯,xn−1∈C⊥, y=y0,y1,⋯,yn−1∈C, then(11)σϱ,λn−1y=λϱy1n−1,λϱy2n−1,⋯,λϱyn−1n−1,ϱy0n−1∈C.
We can get that(12)0=x⋅σϱ,λn−1yλx0ϱy1n−1+λx1ϱy2n−1+⋯+λxn−2ϱyn−1n−1+xn−1ϱy0n−1=λx0ϱy1n−1+x1ϱy2n−1+⋯+xn−2ϱyn−1n−1+λ−1xn−1ϱy0n−1,0=ϱ0=ϱx0ϱy1n−1+x1ϱy2n−1+⋯+xn−2ϱyn−1n−1+λ−1xn−1ϱy0n−1=ϱx0y1+ϱx1y2+⋯+ϱxn−2yn−1+λ−1ϱxn−1y0=σϱ,λ−1x⋅y,so σϱ,λ−1x∈C⊥; hence, C⊥ is a skew θt-λ−1-constacyclic code.
By Lemma2, C⊥ is a skew θt-λ−1-constacyclic code of length n over Rk. By Theorem 1, Cj⊥ is a skew θt-λj−1-constacyclic code over Fq for j=1,2,⋯,k+1.Theorem 3.
LetC=⊕j=1k+1ςjCj be a skew ϱ-λ-constacyclic code of length n over Rk,λ=λ1ς1+λ2ς2+⋯+λk+1ςk+1 is a unit in Rkordϱ∣n,ϱλ=λ. Then, there exists a polynomial ς1g1x+ς2g2x+⋯+ςk+1gk+1x∈Rkx,ϱ subject to C=ς1g1x+ς2g2x+⋯+ςk+1gk+1x, where the right divisor ofxn−λ is ς1g1x+ς2g2x+⋯+ςk+1gk+1x, the generator polynomial of skew θt-λj-constacyclic Cjisgjx∈Fqx,θt, and gjx divides xn−λj on the right for j=1,2,⋯,k+1.Proof.
LetC=⊕j=1k+1ςjCj be a skew ϱ-λ-constacyclic code of length n over Rk. By Theorem 1, Cj is a skew θt-λj-constacyclic code of length n over Fq for j=1,2,⋯,k+1.
Letgjx be the generator polynomial of Cj, then(13)C=ς1g1x,ς2g2x,⋯,ςk+1gk+1x.
LetC′=ς1g1x+ς2g2x+⋯+ςk+1gk+1x. Clearly, C′⊆C.
Becauseςjς1g1x+ς2g2x+⋯+ςk+1gk+1x=ςjgjxforj=1,2,⋯,k+1, so C⊆C′.
Hence,C=C′=ς1g1x+ς2g2x+⋯+ςk+1gk+1x.
Because the right divisor ofxn−λj is gjx for j=1,2,⋯,k+1. Let fjxgjx=xn−λj. Then, ς1f1x+ς2f2x+⋯+ςk+1fk+1xς1g1x+ς2g2x+⋯+ςk+1gk+1x=xn−λ1ς1+λ2ς2+⋯+λk+1ςk+1=xn−λ.
So, the right divisor ofxn−λ is ς1g1x+ς2g2x+⋯+ςk+1gk+1x.Corollary 1.
LetC=⊕j=1k+1ςjCj be a skew ϱ-λ-constacyclic code of length n over Rk,λ=λ1ς1+λ2ς2+⋯+λk+1ςk+1 is a unit in Rk,ordϱ∣n,ϱλ=λ. Then, C⊥=ς1f1∗x+ς2f2∗x+⋯+ςk+1fk+1∗x, ∣C⊥∣=q∑j=1k+1deggjx, where fjxgjx=xn−λj,fjx=a0,j+a1,jx+⋯+an−r,jxn−r∈Fqx,θt,fj∗x=an−r,j+θtan−r−1,jx+⋯+θtn−ra0,jxn−r, and fj∗x is the generate polynomials of skew θt-λj−1-constacyclic Cj⊥ for j=1,2,⋯,k+1.Proof.
LetCj⊥=fj∗xforj=1,2,⋯,k+1, using Theorems 2 and 3, C⊥=⊕j=1k+1ςjCj⊥, then C⊥=∏k+1j=1Cj⊥=q∑k+1j=1deggjx, and we can get that C⊥=ς1f1∗x,ς2f2∗x,⋯,ςk+1fk+1∗x. Let D˜=ς1f1∗x+ς2f2∗x+⋯+ςk+1fk+1∗x. Clearly, D˜⊆C⊥.Because ςjς1f1∗x+ς2f2∗x+⋯+ςk+1fk+1∗x=ςjfj∗xforj=1,2,⋯,k+1, so C⊥⊆D˜.
Therefore,C⊥=D˜=ς1f1∗x+ς2f2∗x+⋯+ςk+1fk+1∗x.


## 4. Quantum Codes from Skewϱ-λ-Constacyclic Codes over Rk

Theorem 4.
LetC=⊕j=1k+1ςjCj be a linear code of length n over Rk, with order C=ql, and the minimum Gray distance of C is dG. Then, ϕkC is a k+1n,l,dG linear code and ϕkC⊥=ϕkC⊥. If C is a self-dual code over Rk, then ϕkC is a self-dual code over Fq.Proof.
By the definition ofϕk, we can have that ϕkC is a k+1n,l,dG linear code.
Leta=a0,a1,⋯,an−1∈C,b=b0,b1,⋯,bn−1∈C⊥,aj=a1,jς1+a2,jς2+⋯+ak+1,jςk+1,bj=b1,jς1+b2,jς2+⋯+bk+1,jςk+1∈Rk, j=0,1,2,⋯,n−1, and ai=ai,0,ai,1,⋯,ai,n−1,bi=bi,0,bi,1,⋯,bi,n−1, i=1,2,…,k+1.Then, a⋅b=∑n−1j=0ajbj=∑n−1j=0∑k+1i=1ai,jbi,jςi=∑k+1i=1aibiTςi=0.So aibiT=0 and i=1,2,⋯,k+1.Since ϕka=a1,a2,⋯,ak+1andϕkb=b1,b2,⋯,bk+1,(14)ϕka⋅ϕkb=ϕkaϕkbT=∑i=1k+1aibiT=0.
So, we haveϕkC⊥⊆ϕkC⊥.
Becauseϕk is bijective, C=ϕkC. Then, ∣ϕkC⊥∣=qk+1n/C=qk+1n/∣ϕkC∣=∣ϕkC⊥∣. We have ϕkC⊥=ϕkC⊥.
IfC is a self-dual code, C=C⊥and then ϕkC⊥=ϕkC⊥=ϕkC.
Therefore,ϕkC is a self-dual code over Fq.Lemma 4.
LetC be a skew θt-λ-constacyclic code of length n over Fq, whose generator polynomial is gx and ordθtn. Then, C contains its dual code if and only if xn−λ is the right divisor of f∗xfx, where λ=±1 and the generator polynomial of C⊥isf∗x.Proof.
LetC⊥=f∗x, where fxgx=xn−λ and λ=±1,C contains its dual code if and only if there exists hx∈Fqx,θt subject to f∗x=hxgx, by Lemma 1, f∗xfx=hxgxfx=hxfxgx=hxxn−λ if and only if the right divisor of f∗xfx is xn−λ.
In the present section, we construct quantum codes from skewϱ-λ-constacyclic over Rk by using the CSS construction [1, 2].Theorem 5.
(CSS Construction). LetC=n,k,dq be a linear codes over Fq, ifC⊥⊆C, then there exists a quantum code n,2k−n,dq.Theorem 6.
LetC=⊕j=1k+1ςjCj be a skew ϱ-λ-constacyclic code of length n over Rk, ordϱ∣n, ϱλ=λ, λ=λ1ς1+λ2ς2+⋯+λk+1ςk+1 is a unit in Rk. Then, C⊥⊆C if and only if the right divisor of fi∗xfix is xn−λi, λi=±1 and i=1,2,⋯,k+1.Proof.
Suppose the right divisor offi∗xfix is xn−λi, by Lemma 4, Ci⊥⊆Ci, i=1,2,⋯,k+1, then ςiCi⊥⊆ςiCi, which implies C⊥=⊕j=1k+1ςjCj⊥⊆⊕j=1k+1ςjCj=C.
On the contrary, letC⊥⊆C, then C⊥=⊕j=1k+1ςjCj⊥⊆⊕j=1k+1ςjCj=C. Hence, Ci⊥⊆Ci. By Lemma 4, we have the right divisor of fi∗xfix as xn−λi,λi=±1,i=1,2,⋯,k+1.
Using Lemma4 and Theorem 6, we can get the following corollary.Corollary 2.
LetC=⊕j=1k+1ςjCj be a skew ϱ-λ-constacyclic code of length n over Rk, where ordϱ∣n,ϱλ=λ, and λ=λ1ς1+λ2ς2+⋯+λk+1ςk+1 is a unit in Rk. Then, C⊥⊆C if and only if Cj⊥⊆Cj,j=1,2,⋯,k+1.Theorem 7.
LetC=⊕j=1k+1ςjCj be a skew ϱ- λ-constacyclic code of length n over Rk, where ordϱ∣n, ϱλ=λ, and λ=λ1ς1+λ2ς2+⋯+λk+1ςk+1 is a unit in Rk. If Cj is a skew θt-λj-constacyclic code over Fq and Cj⊥⊆Cj, where λi=±1 and j=1,2,⋯,k+1, then ϕkC⊥⊆ϕkC and there exists a quantum code k+1n,2l−k+1n,dGq, where the minimum Gray weight of C is dG and the dimension of ϕkC is l.Proof.
SinceCj is a skew θt-λj-constacyclic code over Fq and Cj⊥⊆Cj,λj=±1,j=1,2,⋯,k+1, using Corollary 2, and C⊥⊆C. So, ϕkC⊥⊆ϕkC, by Theorem 4ϕkC⊥=ϕkC⊥. Therefore, ϕkC⊥⊆ϕkC, and by Theorem 4, ϕkC = k+1n,l,dG. Using Theorem 5, there exists a quantum code k+1n,2l−k+1n,dGq.Example 1.
Letn=3 and R2=F27u1,u2/u12=u1,u22=u2,u1u2=u2u1=0, ς1=u1,ς2=u2,ς3=1−u1−u2,∀a∈F27, and θt:F27⟶F27 is defined by θta=a3, and ∀a0+a1u1+a2u2∈R2,ϱa0+a1u1+a2u2=a0θt+a1θtu1+a2θtu2. Then, ϱ=3,ordϱn.(15)x3+1=ω24+ω7x+ω8x2ω2x+ω2∈F27x,θt,x3−1=ω12+ω22x+x2x+ω∈F27x,θt.
LetC be a skew ϱ-ς1+ς2+−1ς3-constacyclic code of length 3 over R2. Let gx=ς1g1x+ς2g2x+ς3g3x, where g1x=g2x=ω2x+ω2,g3x=x+ω. Then, C1=g1xandC2=g2x are skew negacyclic codes of length 3 over F27.C3=g3x is a skew cyclic code of length 3 over F27. By Theorem 4, ϕ2C=9,6,327. Using Theorem 7, C⊥⊆C. So, we can get a quantum code 9,3,327 such that n−k+2−2d=2.Example 2.
Letn=8 and R3=F9u1,u2,u3/ui2=−ui,uiuj=ujui=0,ς1=−u1,ς2=−u2,ς3=−u3,ς4=1+u1+u2+u3,∀a∈F9 and θt:F9⟶F9 is defined by θta=a3, and ∀a0+a1u1+a2u2+a3u3∈R3andϱa0+a1u1+a2u2+a3u3=a0θt+a1θtu1+a2θtu2+a3θtu3. Then, ϱ=2,ordϱn.(16)x8−1=ω2x5+ω2x4+ωx3+ωx2+2x+2ω2x3+ω2x2+2x+1∈F9x,θt,x8+1=ω7x6+ω3x5+ω5x4+ω7x3+ω6x2+ω7x+1ωx2+ω3x+1∈F9x,θt.
LetC be a skew ϱ-ς1+ς2+−1ς3+−1ς4-constacyclic code of length 8 over R3. Let gx=ς1g1x+ς2g2x+ς3g3x+ς4g4x, where g1x=g2x=ω2x3+ω2x2+2x+1,g3x=g4x=ωx2+ω3x+1. Then, C1=g1x and C2=g2x are skew cyclic codes of length 8 over F9.C3=g3x and C4=g4x are skew negacyclic codes of length 8 over F9. By Theorem 4, ϕ3C=32,22,49. By Theorem 7, C⊥⊆C. So, we can get a quantum code 32,12,49.Example 3.
Letn=12 and R3=F9u1,u2,u3/ui2=−ui,uiuj=ujui=0,ς1=−u1,ς2=−u2,ς3=−u3,ς4=1+u1+u2+u3,∀a∈F9, and θt:F9⟶F9 is defined by θta=a3, and ∀a0+a1u1+a2u2+a3u3∈R3andϱa0+a1u1+a2u2+a3u3=a0θt+a1θtu1+a2θtu2+a3θtu3. Then, ϱ=2,ordϱn.(17)x12−1=ω2x10+ω2x8+ω2x6+ω2x4+ω2x2+ω2ω6x2+ω2∈F9x,θt,x12+1=x10+ω6x8+2x6+ω2x4+x2+ω6x2+ω2∈F9x,θt.
LetC be a skew ϱ-ς1+−1ς2+ς3+−1ς4-constacyclic code of length 12 over R3. Let gx=ς1g1x+ς2g2x+ς3g3x+ς4g4x, where g1x=g3x=ω6x2+ω2andg2x=g4x=x2+ω2. Then, C1=g1xandC3=g3x are skew cyclic codes of length 12 over F9, and C2=g2xandC4=g4x are skew negacyclic codes of length 12 over F9. By Theorem 4, ϕ3C=48,40,39. By Theorem 7, C⊥⊆C. So, we can get a quantum code 48,32,39, which has larger dimension than 48,24,39 in [21].
In Table1, some new quantum codes are given from skew ϱ-λ constacyclic over Rk. Our quantum codes 18,12,327,9,3,327,18,12,347,18,12,3169,28,22,349 have the parameters such that n−k−2d+2=2. These codes are approached quantum MDS codes (satisfying quantum singleton bound n−k−2d+2=0). Moreover, our obtained quantum codes 48,32,39,40,30,325,56,46,349 have larger dimensions than the quantum codes 48,24,39,40,24,325,56,40,349 in [21].Table 1
Quantum codes from skewϱ-λ-constacyclic over Rk.
qnkλ1,⋯,λk+1<g1x,⋯,gk+1x>ϕkCn,l,dqn′,k′,d′q49621,−1,−131,ω1,ω118,15,318,12,349n−k−2d+2=027321,1,−11ω,1ω,ω2ω29,6,39,3,327n−k−2d+2=049621,−1,−131,ω1,ω118,15,318,12,349n−k−2d+2=0169621,−1,1ω161,ω381,ω16118,15,328,22,349n−k−2d+2=0491411,−11ω221,ω3128,25,348,32,39n−k−2d+2=091231,−1,1,−1ω60ω2,10ω2,ω60ω2,10ω248,40,348,32,3948,32,39 [21]251031,1,−1,−1ω8ω41,ω41,ω101,ω10140,35,340,30,32540,24,325 [21]491431,−1,−1,11ω221,ω31,ω31,ω6156,51,356,46,34956,46,349 [21]


## 5. Conclusions

In this article, we construct quantum codes by studying the structure of skewϱ-λ-constacyclic codes over a finite nonchain ring Rk=Fqu1,u2,⋯,uk/ui2=αiui,uiuj=ujui=0, where q=pm,p is an odd prime, and αi is a unit over Fq,i,j=1,2,⋯,k. The major contributions are as follows: we study the structure of skew ϱ-λ-constacyclic code of length n over Rk and give the necessary and sufficient conditions of dual-containing skew constacyclic codes. Our results will enrich the code source of quantum codes. Besides, we obtain some new quantum codes from skew ϱ-λ-constacyclic over Rk by using the CSS construction. Our obtained quantum codes are approached quantum MDS codes or have larger dimensions than [21].



---

*Source: 8196228-2023-03-22.xml*