Non-binary quantum codes from constacyclic codes over 𝔽q[u1, u2,…,uk]/⟨ui3 = ui, uiuj = ujui⟩

Bo Kong; Xiying Zheng

doi:10.1515/math-2022-0459

Open Access Published by De Gruyter Open Access September 12, 2022

Non-binary quantum codes from constacyclic codes over 𝔽_q[u₁, u₂,…,u_k]/⟨u_i³ = u_i, u_iu_j = u_ju_i⟩

Bo Kong and Xiying Zheng

From the journal Open Mathematics

https://doi.org/10.1515/math-2022-0459

Abstract

Let q = p m , p be an odd prime, and R k = F q [ u 1 , u 2 , … , u k ] / ⟨ u i 3 = u i , u i u j = u j u i ⟩ , where k ≥ 1 and 1 ≤ i , j ≤ k . In this article, we define a Gray map from R k n to F q 3 k n . We study constacyclic codes over R k and construct non-binary quantum codes over F q .

Keywords: constacyclic codes; quantum codes; Gray map; dual-containing codes

MSC 2010: 94B05; 94B15

1 Introduction

Recently, constructing quantum error-correcting codes has important significance in theory and practice. Calderbank et al. [1] gave a way to construct quantum error correcting codes from classical error correcting codes. Constacyclic codes that have good error-correcting properties are an important class of linear codes. Constacyclic codes also have rich algebraic structures that can be encoded with shift registers. Due to their rich algebraic structure, constacyclic codes over finite fields have been studied by many authors [2,3,4], and many good quantum codes have been constructed by using classical cyclic and constacyclic codes over finite fields [5,6, 7,8].

In recent years, there are a lot of works about constacyclic codes over finite rings of the form F p m + u F p m + ⋯ + u e − 1 F p m by many authors [9,10, 11,12], where u e = 0 . The class of finite commutative rings of the form R + u R has been studied by many authors [13,14], where u 2 = 1 . The class of finite commutative rings of the form F p m [ u 1 , u 2 , … , u k ] / ⟨ u i 2 = u i , u i u j = u j u i ⟩ has been studied by many authors [15,16, 17,18], where u i 2 = u i . Due to their rich algebraic structure, many good quantum codes have been constructed by using classical cyclic and negacyclic codes, and there are a few quantum codes constructed by using constacyclic codes over finite rings. Dertli and Cengellenmis [19] constructed quantum codes from constacyclic codes over ring F p + u F p + v F p with u 2 = u , v 2 = v , and u v = v u = 0 . Wang et al. [20] constructed non-binary quantum codes from ( 1 − 2 v ) -constacyclic codes over F q 2 + v F q 2 with v 2 = v . Gowdhaman et al. [21] constructed quantum codes from λ -constacyclic codes over the ring F p [ u , v ] ⟨ v 3 − v , u 3 − u , u v − v u ⟩ . Li et al. [22] constructed quantum error correcting codes by Hermitian construction and obtained some good quantum codes. In [23], quantum codes from cyclic codes over F 2 + u F 2 + v F 2 + u v F 2 for arbitrary length n were constructed. In [24], the structure of cyclic codes over the ring F q + v 1 F q + ⋯ + v r F q was studied, and quantum codes from cyclic codes were constructed. Furthermore, some new non-binary quantum codes were obtained. In [25], quantum codes over F p were constructed by using the cyclic codes of length n over F p [ u , v , w ] / ⟨ u 2 − 1 , v 2 − 1 , w 2 − 1 , u v − v u , v w − w v , w u − u w ⟩ . In [26], some non-binary quantum codes were obtained from ( 1 − 2 v ) -constacyclic codes over the finite non-chain ring F q + u F q + v F q + u v F q . Grassl et al. [27] presented families of non-binary quantum codes which were optimal in the sense that the minimum distance was maximal.

The purpose of this article is to continue this line of research. First, we determine the algebraic structures of all λ -constacyclic codes of R k = F q [ u 1 , u 2 , … , u k ] / ⟨ u i 3 = u i , u i u j = u j u i ⟩ . Second, we construct quantum codes from constacyclic codes over R k .

The rest of this article is arranged as follows: In Section 2, we give some results of R k and the definition of the Gray map from R k n to F q 3 k n . In Section 3, we discuss the algebraic structure of constacyclic codes over R k . In Section 4, we give the parameters of quantum error correcting codes from constacyclic codes over R k .

2 Preliminaries

Let F q be a finite field with q elements, where p is an odd prime and q = p m , and let

R k = F q [ u 1 , u 2 , … , u k ] / ⟨ u i 3 = u i , u i u j = u j u i ⟩ .

Clearly, R k is a Frobenius ring but not local, and R k has cardinality q ( 3 k ) .

Lemma 2.1

Let I = ⟨ w 1 , w 2 , … , w k ⟩ , where w i ∈ 1 − u i 2 , u i 2 + u i 2 , u i 2 − u i 2 , then I is an ideal of R k , and the cardinality of I is q ( 3 k − 1 ) , and the number of such ideals is 3 k .

Proof

The elements of I are of the form

∑ a i 1 i 2 ⋯ i k w 1 i 1 w 2 i 2 ⋯ w k i k ∣ a i 1 i 2 ⋯ i k ∈ F q , i s ∈ { 0 , 1 , 2 } , s = 1 , 2 , … , k .

It is easy to see that there are 3 k − 1 choices of w 1 i 1 w 2 i 2 ⋯ w k i k . So, I has cardinality q ( 3 k − 1 ) . It is easy to see that the other ideals are isomorphic to I , for each w i has three choices, so the number of such ideals is 3 k .□

Let ϖ i = ⟨ w i 1 , w i 2 , … , w i k ⟩ be an ideal in Lemma 2.1, where w i j ∈ 1 − u j 2 , u j 2 + u j 2 , u j 2 − u j 2 , 1 ≤ i ≤ 3 k , 1 ≤ j ≤ k .

Let ς i = w i 1 w i 2 ⋯ w i k , where w i j ∈ 1 − u j 2 , u j 2 + u j 2 , u j 2 − u j 2 , i = 1 , 2 , … , 3 k , j = 1 , 2 , … , k . We can have that

( 1 − u j 2 ) 2 = 1 − u j 2 , u j 2 + u j 2 2 = u j 2 + u j 2 , u j 2 − u j 2 2 = u j 2 − u j 2 , ( 1 − u j 2 ) u j 2 + u j 2 = 0 , ( 1 − u j 2 ) u j 2 − u j 2 = 0 , u j 2 − u j 2 u j 2 + u j 2 = 0 ,

ς i ς j = 0 , when i ≠ j , and ς i 2 = ς i , when i = 1 , 2 , … , 3 k .

By the induction method over R k , we can obtain that 1 = ς 1 + ς 2 + ⋯ + ς 3 k and R k = ς 1 R k ⊕ ς 2 R k ⊕ ⋯ ⊕ ς 3 k R k . ∀ r ∈ R k , then r can be expressed uniquely as the form r = r 1 ς 1 + r 2 ς 2 + ⋯ + r 3 k ς 3 k , where r i ∈ F q , i = 1 , 2 , … , 3 k .

By the same method of Theorem 2.3 in [18], we have the following theorem.

Theorem 2.1

ϖ 1 , ϖ 2 , … , ϖ 3 k are maximal ideals of R k , and R k ≅ F q 3 k .

By the aforementioned theorem, it can be easily seen that R k is a principal ideal ring, not a chain ring.

For a = a 1 ς 1 + a 2 ς 2 + ⋯ + a 3 k ς 3 k ∈ R k , we can define ϕ k : R k → F q 3 k by a ↦ ( a 1 , a 2 , … , a 3 k ) , and we expand ϕ k as

ϕ k : R k n → F q 3 k n ,

( a 0 , a 1 , … , a n − 1 ) ↦ ( a 1 , 0 , … , a 1 , n − 1 , a 2 , 0 , … , a 2 , n − 1 , … , a 3 k , 0 , … , a 3 k , n − 1 ) ,

where a i = a 1 , i ς 1 + a 2 , i ς 2 + ⋯ + a 3 k , i ς 3 k ∈ R k , i = 1 , 2 , … , 3 k .

Let R k n be the R k -submodule, if C is an R k -submodule of R k n , then C is a linear code of length n over R k . Every codeword c = ( c 0 , c 1 , … , c n − 1 ) ∈ C can be represented as

c = ( c 0 , c 1 , … , c n − 1 ) ↔ c ( x ) = ∑ i = 0 n − 1 c i x i ∈ R k [ x ] .

If C is invariant under constacyclic shift operator σ λ : R k n → R k n by

σ λ ( c 0 , c 1 , … , c n − 1 ) = ( λ c n − 1 , c 0 , … , c n − 2 ) ,

then C is called a λ -constacyclic code of length n over R k .

Let x = ( x 0 , x 1 , … , x n − 1 ) and y = ( y 0 , y 1 , … , y n − 1 ) ∈ R k n . The Euclidean inner product of x and y is defined by x ⋅ y = ∑ i = 0 n − 1 x i y i . If x ⋅ y = 0 , then x and y are orthogonal.

If C is a linear code, the Euclidean dual code C ⊥ = { x ∣ ∀ y ∈ C , x ⋅ y = 0 } is a linear code too. A code C is Euclidean self-orthogonal if C ⊆ C ⊥ , and Euclidean self-dual if C = C ⊥ .

Let w H ( ϕ k ( r ) ) denote the Hamming weight of the image of r under ϕ k , ∀ r ∈ R k , the Lee weight of r is defined as w L ( r ) = w H ( ϕ k ( r ) ) .

∀ r = ( x 1 , x 2 , … , x n ) ∈ R k n , the Lee weight of r is defined as w L ( r ) = ∑ i = 1 n w L ( x i ) , the Lee distance of codewords x , y over R k n is defined as d L ( x , y ) = w L ( x − y ) , and the Lee distance of C is defined as

d L ( C ) = min { d L ( x − y ) , x , y ∈ C , x ≠ y } .

By the definition above, it is easy to see that ϕ k is both a distance preserving map and a linear map from R k n to F q 3 k n .

3 Constacyclic codes over R k

∀ x = ∑ j = 1 3 k x j ς j , y = ∑ j = 1 3 k y j ς j ∈ R k n , where x j = ( x 0 j , x 1 j , … , x n − 1 , j ) ∈ F q n , y j = ( y 0 j , y 1 j , … , y n − 1 , j ) ∈ F q n , we can have that

x ⋅ y = ∑ j = 1 3 k ( x j ⋅ y j ) ς j .

Let C be a linear code of length n over R k . Let

C j = x j ∈ F q n ∣ ∑ i = 1 3 k x i ς i ∈ C , x i ∈ F q n , , j = 1 , 2 , … , 3 k .

Clearly, C 1 , C 2 , … , C 3 k are linear codes of length n over F q , and C = ⊕ j = 1 3 k ς j C j , ∣ C ∣ = ∏ j = 1 3 k ∣ C j ∣ .

Lemma 3.1

Let ( λ 1 ς 1 + λ 2 ς 2 + ⋯ + λ 3 k ς 3 k ) be an element over R k . Then ( λ 1 ς 1 + λ 2 ς 2 + ⋯ + λ 3 k ς 3 k ) is a unit over R k if and only if λ 1 , λ 2 , … , λ 3 k are units over F q .

Proof

Let ( λ 1 ς 1 + λ 2 ς 2 + ⋯ + λ 3 k ς 3 k ) be a unit of R k , then there exist β 1 , β 2 , … , β 3 k ∈ F q such that

( λ 1 ς 1 + λ 2 ς 2 + ⋯ + λ 3 k ς 3 k ) ( β 1 ς 1 + β 2 ς 2 + ⋯ + β 3 k ς 3 k ) = 1 .

Then ( λ 1 β 1 ς 1 + λ 2 β 2 ς 2 + ⋯ + λ 3 k β 3 k ς 3 k ) = 1 , which implies ( λ 1 β 1 ς 1 + λ 2 β 2 ς 2 + ⋯ + λ 3 k β 3 k ς 3 k ) ς i = λ i β i ς i = ς i , so λ i β i = 1 , where i = 1 , 2 , … , 3 k . So λ 1 , λ 2 , … , λ 3 k are units over F q .

Conversely, if λ 1 , λ 2 , … , λ 3 k are units over F q , then there exist β 1 , β 2 , … , β 3 k ∈ F q such that λ i β i = 1 , where i = 1 , 2 , … , 3 k . We can obtain that ( λ 1 ς 1 + λ 2 ς 2 + ⋯ + λ 3 k ς 3 k ) ( β 1 ς 1 + β 2 ς 2 + ⋯ + β 3 k ς 3 k ) = λ 1 β 1 ς 1 + λ 2 β 2 ς 2 + ⋯ + λ 3 k β 3 k ς 3 k = ς 1 + ς 2 + ⋯ + ς 3 k = 1 . So ( λ 1 ς 1 + λ 2 ς 2 + ⋯ + λ 3 k ς 3 k ) is a unit over R k .□

Theorem 3.1

Let C = ⊕ j = 1 3 k ς j C j be a linear code of length n over R k , then C is a ( λ 1 ς 1 + λ 2 ς 2 + ⋯ + λ 3 k ς 3 k ) -constacyclic code over R k if and only if C i is λ i -constacyclic codes over F q , where i = 1 , 2 , … , 3 k , and λ 1 ς 1 + λ 2 ς 2 + ⋯ + λ 3 k ς 3 k is a unit of R k .

Proof

This can be proved by the same method of Theorem 4.1 in [18].□

Theorem 3.2

Let C = ⊕ j = 1 3 k ς j C j be a ( λ 1 ς 1 + λ 2 ς 2 + ⋯ + λ 3 k ς 3 k ) -constacyclic code of length n over R k , then C = ⟨ ς 1 g 1 ( x ) + ς 2 g 2 ( x ) + ⋯ + ς 3 k g 3 k ( x ) ⟩ , where g i is the generator polynomial of C i , i = 1 , 2 , … , 3 k .

Proof

This can be proved by the same method of Theorem 4.2 in [18].□

Theorem 3.3

Let C = ⟨ ς 1 g 1 ( x ) + ς 2 g 2 ( x ) + ⋯ + ς 3 k g 3 k ( x ) ⟩ be a ( λ 1 ς 1 + λ 2 ς 2 + ⋯ + λ 3 k ς 3 k ) -constacyclic code of length n over R k . Then C ⊥ = ⟨ ς 1 f 1 ∗ ( x ) + ς 2 f 2 ∗ ( x ) + ⋯ + ς 3 k f 3 k ∗ ( x ) ⟩ is a ( λ 1 − 1 ς 1 + λ 2 − 1 ς 2 + ⋯ + λ 3 k − 1 ς 3 k ) -constacyclic code of length n over R k , ∣ C ⊥ ∣ = q ∑ i = 1 3 k deg ( g i ) , where f i ∗ ( x ) is the reciprocal polynomial of f i ( x ) , i.e., f i ( x ) = ( x n − λ i ) / g i ( x ) , f i ∗ ( x ) = x deg ( f i ) f i ( x − 1 ) , for i = 1 , 2 , … , 3 k .

Proof

Let C = ⟨ ς 1 g 1 ( x ) + ς 2 g 2 ( x ) + ⋯ + ς 3 k g 3 k ( x ) ⟩ be a λ -constacyclic code of length n over R k , where λ = λ 1 ς 1 + λ 2 ς 2 + ⋯ + λ 3 k ς 3 k .

∀ x = ( x 0 , x 1 , … , x n − 1 ) ∈ C ⊥ , ∀ y = ( y 0 , y 1 , … , y n − 1 ) ∈ C , then σ λ n − 1 ( y ) = ( λ y 1 , λ y 2 , … , λ y n − 1 , y 0 ) ∈ C , we can obtain that

0 = x ⋅ σ λ n − 1 ( y ) = λ x 0 y 1 + λ x 1 y 2 + ⋯ + λ x n − 2 y n − 1 + x n − 1 y 0 = λ ( x 0 y 1 + x 1 y 2 + ⋯ + x n − 2 y n − 1 + λ − 1 x n − 1 y 0 ) = λ σ λ − 1 ( x ) ⋅ y .

We have σ λ − 1 ( x ) ∈ C ⊥ , so C ⊥ is a λ − 1 -constacyclic code.

By Lemma 3.1, λ − 1 = λ 1 − 1 ς 1 + λ 2 − 1 ς 2 + ⋯ + λ 3 k − 1 ς 3 k , so C ⊥ is a ( λ 1 − 1 ς 1 + λ 2 − 1 ς 2 + ⋯ + λ 3 k − 1 ς 3 k ) -constacyclic code of length n over R k .

Let D i = ⟨ f i ∗ ( x ) ⟩ , where f i ∗ ( x ) is the reciprocal polynomial of f i ( x ) , i.e., f i ( x ) = ( x n − λ i ) / g i ( x ) , f i ∗ ( x ) = x deg ( f i ) f i ( x − 1 ) , g i is the generator polynomial of C i , i = 1 , 2 , … , 3 k .

Let D = ⊕ j = 1 3 k ς j D j , then ∣ D ∣ = ∏ j = 1 3 k ∣ D j ∣ = q ( ∑ i = 1 3 k deg ( g i ) ) , and it follows that D has the form

D = ⟨ ς 1 f 1 ∗ ( x ) , ς 2 f 2 ∗ ( x ) , … , ς 3 k f 3 k ∗ ( x ) ⟩ .

Let D ˜ = ⟨ ς 1 f 1 ∗ ( x ) + ς 2 f 2 ∗ ( x ) + ⋯ + ς 3 k f 3 k ∗ ( x ) ⟩ . We can have that D ˜ ⊆ D .

Note that

ς i [ ς 1 f 1 ∗ ( x ) + ς 2 f 2 ∗ ( x ) + ⋯ + ς 3 k f 3 k ∗ ( x ) ] = ς i f i ∗ ( x ) ,

where i = 1 , 2 , … , 3 k .

We can obtain that D ⊆ D ˜ . So, D ˜ = D , and D = ⟨ ς 1 f 1 ∗ ( x ) + ς 2 f 2 ∗ ( x ) + ⋯ + ς 3 k f 3 k ∗ ( x ) ⟩ .

Note that

( ς 1 f 1 ∗ ( x ) + ς 2 f 2 ∗ ( x ) + ⋯ + ς 3 k f 3 k ∗ ( x ) ) ( ς 1 g 1 ∗ ( x ) + ς 2 g 2 ∗ ( x ) + ⋯ + ς 3 k g 3 k ∗ ( x ) ) = ς 1 f 1 ∗ ( x ) g 1 ∗ ( x ) + ς 2 f 2 ∗ ( x ) g 2 ∗ ( x ) + ⋯ + ς 3 k f 3 k ∗ ( x ) g 3 k ∗ ( x ) = ς 1 ( 1 − x n λ 1 ) + ς 2 ( 1 − x n λ 2 ) + ⋯ + ς 3 k ( 1 − x n λ 3 k ) = 1 − x n ( λ 1 ς 1 + λ 2 ς 2 + ⋯ + λ 3 k ς 3 k ) = − ( x n − ( λ 1 − 1 ς 1 + λ 2 − 1 ς 2 + ⋯ + λ 3 k − 1 ς 3 k ) ) ( λ 1 ς 1 + λ 2 ς 2 + ⋯ + λ 3 k ς 3 k ) .

So, ( ς 1 f 1 ∗ ( x ) + ς 2 f 2 ∗ ( x ) + ⋯ + ς 3 k f 3 k ∗ ( x ) ) ∣ ( x n − ( λ 1 − 1 ς 1 + λ 2 − 1 ς 2 + ⋯ + λ 3 k − 1 ς 3 k ) ) , we can obtain that D ⊆ C ⊥ .

For R k is a Frobenius ring, ∣ C ∣ ∣ C ⊥ ∣ = ∣ R k ∣ n , hence ∣ C ⊥ ∣ = ∣ R k ∣ n ∣ C ∣ = q ( ∑ i = 1 3 k deg ( g i ) ) = ∣ D ∣ .

So, C ⊥ = D = ⟨ ς 1 f 1 ∗ ( x ) + ς 2 f 2 ∗ ( x ) + ⋯ + ς 3 k f 3 k ∗ ( x ) ⟩ .□

4 Quantum codes from constacyclic codes over R k

Theorem 4.1

Let C = ⊕ j = 1 3 k ς j C j be a linear code of length n over R k , then C ⊥ = ∑ j = 1 3 k ς j C j ⊥ , where C j ⊥ is a Euclidean dual code of C j , where j = 1 , 2 , … , 3 k .

Proof

Let C ˜ = ⊕ j = 1 3 k ς j C j ⊥ . ∀ x = ∑ j = 1 3 k ς j x j ∈ C , x ˜ = ∑ j = 1 3 k ς j x j ˜ ∈ C ˜ , x ⋅ x ˜ = ∑ j = 1 3 k ( x j x j ˜ ) ς j , where x j ∈ C j , and x j ˜ ∈ C j ⊥ . We can have x ⋅ x ˜ = 0 , so C ˜ ⊆ C ⊥ .

For R k is a Frobenius ring, ∣ C ∣ ∣ C ⊥ ∣ = ∣ R k ∣ n . Hence,

∣ C ˜ ∣ = ∏ j = 1 3 k ∣ C j ⊥ ∣ = ∏ j = 1 3 k p n ∣ C j ∣ = ∣ R k ∣ n ∣ C ∣ = ∣ C ⊥ ∣ .

It follows that

C ⊥ = C ˜ . □

Theorem 4.2

Let C = ⊕ j = 1 3 k ς j C j be a linear code of length n over R k , then C is a Euclidean self-orthogonal code over R k if and only if C j is a Euclidean self-orthogonal code over F q , where j = 1 , 2 , … , 3 k .

Proof

By Theorem 4.1, C ⊆ C ⊥ if and only if C j ⊆ C j ⊥ , so if C is a Euclidean self-orthogonal code over R k , then C j is a Euclidean self-orthogonal code over F q , where j = 1 , 2 , … , 3 k .□

Let C be a linear code of length n over R k , for any c = c 1 ς 1 + c 2 ς 2 + ⋯ + c 3 k ς 3 k ∈ C , and ϕ k ( c ) = ( c 1 , c 2 , … , c 3 k ) ∈ F q 3 k n .

Theorem 4.3

Let C = ⊕ j = 1 3 k ς j C j be a linear code of length n over R k with ∣ C ∣ = q l and the minimum Lee distance d L ( C ) = d . Then ϕ k ( C ) is a linear code with parameter [ 3 k n , l , d ] . If C is a Euclidean self-orthogonal code over R k , then ϕ k ( C ) is a Euclidean self-orthogonal code over F q .

Proof

By the definition of Gray map ϕ k , we can know that ϕ k ( C ) is a linear code with parameter [ 3 k n , l , d ] .

Let C be a Euclidean self-orthogonal code, x = ∑ j = 1 3 k ς j x j ∈ C , and y = ∑ j = 1 3 k ς j y j ∈ C , where x j , y j ∈ F q n , then x ⋅ y = ∑ j = 1 3 k ς j x j y j = 0 , which implies that x j y j = 0 , so

ϕ k ( x ) ⋅ ϕ k ( y ) = ∑ j = 1 3 k x j y j = 0 .

So ϕ k ( C ) is a Euclidean self-orthogonal code over F q .□

Lemma 4.1

[26] Let C be a constacyclic code with generator polynomial g ( x ) over F q . Then, C contains its dual code if and only if x n − λ ≡ 0 ( mod g ( x ) g ∗ ( x ) ) , where g ∗ ( x ) is the reciprocal polynomial of g ( x ) , λ = ± 1 .

Theorem 4.4

[27] Let C 1 and C 2 be [ n , k 1 , d 1 ] q and [ n , k 2 , d 2 ] q linear codes over F q , with C 2 ⊥ ⊆ C 1 ⊥ . Furthermore, let d = min ( d 1 , d 2 ) . Then, there exists a quantum error-correcting code C with parameters C = [ n , k 1 + k 2 − n , ≥ d ] q . In particular, if C 1 ⊥ ⊆ C 1 , then there exists a quantum error-correcting code C with parameters C = [ n , 2 k 1 − n , ≥ d 1 ] .

Theorem 4.5

Let C = ⊕ j = 1 3 k ς j C j be a ( λ 1 ς 1 + λ 2 ς 2 + ⋯ + λ 3 k ς 3 k ) -constacyclic code over R k , where g i ( x ) is the generator of polynomial of C i . Then C ⊥ ⊆ C if and only if C i ⊥ ⊆ C i for i = 1 , 2 , … , 3 k .

Proof

If C i ⊥ ⊆ C i for i = 1 , 2 , … , 3 k , then ς i C i ⊥ ⊆ ς i C i , ς 1 C 1 ⊥ ⊕ ς 2 C 2 ⊥ ⊕ ⋯ ⊕ ς 3 k C 3 k ⊥ ⊆ ς 1 C 1 ⊕ ς 2 C 2 ⊕ ⋯ ⊕ ς 3 k C 3 k . So C ⊥ ⊆ C . Conversely, if C ⊥ ⊆ C , then, ς 1 C 1 ⊥ ⊕ ς 2 C 2 ⊥ ⊕ ⋯ ⊕ ς 3 k C 3 k ⊥ ⊆ ς 1 C 1 ⊕ ς 2 C 2 ⊕ ⋯ ⊕ ς 3 k C 3 k , so, C i ⊥ ⊆ C i for i = 1 , 2 , … , 3 k .□

According to Lemma 4.1 and Theorem 4.5, we obtain the following corollary directly.

Corollary 4.1

Let C = ⊕ j = 1 3 k ς j C j be a ( λ 1 ς 1 + λ 2 ς 2 + ⋯ + λ 3 k ς 3 k ) -constacyclic code over R k . Then C ⊥ ⊆ C if and only if C i is λ i -constacyclic codes over F q , where λ i = ± 1 , i = 1 , 2 , … , 3 k .

Using Theorems 4.3–4.5, we can construct quantum codes.

Theorem 4.6

Let C = ⊕ j = 1 3 k ς j C j be a ( λ 1 ς 1 + λ 2 ς 2 + ⋯ + λ 3 k ς 3 k ) -constacyclic code over R k . If C i is λ i -constacyclic code over F q , where λ i = ± 1 , i = 1 , 2 , … , 3 k , then C ⊥ ⊆ C and there exists a quantum error-correcting code with parameters [ 3 k n , 2 l − 3 k n , ≥ d L ] q , where d L is the minimum Lee weight of code C and l is the dimension of the linear code ϕ k ( C ) .

Example 4.1

Let n = 30 and R 2 = F 5 [ u 1 , u 2 ] / ⟨ u i 3 = u i , u i u j = u j u i ⟩ , ς 1 = ( 1 − u 1 2 ) ( 1 − u 2 2 ) , ς 2 = ( 1 − u 1 2 ) ( u 2 2 + u 2 ) 2 , ς 3 = ( 1 − u 1 2 ) ( u 2 2 − u 2 ) 2 , ς 4 = ( 1 − u 2 2 ) ( u 1 2 + u 1 ) 2 , ς 5 = ( u 1 2 + u 1 ) ( u 2 2 + u 2 ) 4 , ς 6 = ( u 1 2 + u 1 ) ( u 2 2 − u 2 ) 4 , ς 7 = ( 1 − u 2 2 ) ( u 1 2 − u 1 ) 2 , ς 8 = ( u 1 2 − u 1 ) ( u 2 2 + u 2 ) 4 , ς 9 = ( u 1 2 − u 1 ) ( u 2 2 − u 2 ) 4 .

x 30 + 1 = ( x + 2 ) 5 ( x + 3 ) 5 ( x 2 + 2 x + 4 ) 5 ( x 2 + 3 x + 4 ) 5 , x 30 − 1 = ( x + 1 ) 5 ( x + 4 ) 5 ( x 2 + x + 1 ) 5 ( x 2 + 4 x + 1 ) 5 in F 5 ( x ) .

Let C be a ( 1 − 2 u 1 2 u 2 2 ) -constacyclic code of length 30 over R 2 = F 5 [ u 1 , u 2 ] / ⟨ u i 3 = u i , u i u j = u j u i ⟩ . Let g ( x ) = ς 1 g 1 + ς 2 g 2 + ς 3 g 3 + ς 4 g 4 + ς 5 g 5 + ς 6 g 6 + ς 7 g 7 + ς 8 g 8 + ς 9 g 9 be the generator polynomial of C , where g 1 = g 2 = g 3 = x + 1 , g 4 = g 7 = x + 4 , g 5 = g 6 = x + 2 , and g 8 = g 9 = x + 3 . By Theorem 4.5, we have C ⊥ ⊆ C , ϕ 2 ( C ) is a linear code over F 5 with parameters [ 270 , 261 , 2 ] . By Theorem 4.6, we obtain a quantum error-correcting code with parameters [ 270 , 252 , ≥ 2 ] 5 .

Example 4.2

Let n = 15 and R 2 = F 7 [ u 1 , u 2 ] / ⟨ u i 3 = u i , u i u j = u j u i ⟩ , ς 1 = ( 1 − u 1 2 ) ( 1 − u 2 2 ) , ς 2 = ( 1 − u 1 2 ) ( u 2 2 + u 2 ) 2 , ς 3 = ( 1 − u 1 2 ) ( u 2 2 − u 2 ) 2 , ς 4 = ( 1 − u 2 2 ) ( u 1 2 + u 1 ) 2 , ς 5 = ( u 1 2 + u 1 ) ( u 2 2 + u 2 ) 4 , ς 6 = ( u 1 2 + u 1 ) ( u 2 2 − u 2 ) 4 , ς 7 = ( 1 − u 2 2 ) ( u 1 2 − u 1 ) 2 , ς 8 = ( u 1 2 − u 1 ) ( u 2 2 + u 2 ) 4 , ς 9 = ( u 1 2 − u 1 ) ( u 2 2 − u 2 ) 4 .

x 15 − 1 = ( x + 3 ) ( x + 5 ) ( x + 6 ) ( x 4 + x 3 + x 2 + x + 1 ) ( x 4 + 2 x 3 + 4 x 2 + x + 2 ) ( x 4 + 4 x 3 + 2 x 2 + x + 4 ) , x 15 + 1 = ( x + 1 ) ( x + 2 ) ( x + 4 ) ( x 4 + 3 x 3 + 2 x 2 + 6 x + 4 ) ( x 4 + 5 x 3 + 4 x 2 + 6 x + 2 ) ( x 4 + 6 x 3 + x 2 + 6 x + 1 ) .

Let C be a ( 1 − 2 u 1 2 u 2 2 ) -constacyclic code of length 15 over R 2 = F 7 [ u 1 , u 2 ] / ⟨ u i 3 = u i , u i u j = u j u i ⟩ . Let g 1 = x 4 + x 3 + x 2 + x + 1 , g 2 = g 3 = x 4 + 2 x 3 + 4 x 2 + x + 2 , g 4 = g 7 = x 4 + 4 x 3 + 2 x 2 + x + 4 , g 5 = x 4 + 3 x 3 + 2 x 2 + 6 x + 4 , g 6 = x 4 + 5 x 3 + 4 x 2 + 6 x + 2 , and g 8 = g 9 = x 4 + 6 x 3 + x 2 + 6 x + 1 . By Theorem 4.5, we have C ⊥ ⊆ C , ϕ 2 ( C ) is a linear code over F 7 with parameters [ 135 , 99 , 4 ] . By Theorem 4.6, we obtain a quantum error-correcting code with parameters [ 135 , 63 , ≥ 4 ] 7 .

Example 4.3

Let n = 21 and R 1 = F 7 [ u 1 ] / ⟨ u 1 3 = u 1 ⟩ , ς 1 = ( 1 − u 1 2 ) , ς 2 = ( u 1 2 − u 1 ) 2 , ς 3 = ( u 1 2 + u 1 ) 2 .

x 21 − 1 = ( x + 3 ) 7 ( x + 5 ) 7 ( x + 6 ) 7 , x 21 + 1 = ( x + 1 ) 7 ( x + 2 ) 7 ( x + 4 ) 7 .

Let C be a ( 1 − 2 u 1 2 ) -constacyclic code of length 15 over R 1 . g 1 = x + 3 , g 2 = x + 2 , g 3 = x + 4 . By Theorem 4.5, we have C ⊥ ⊆ C , ϕ 1 ( C ) is a linear code over F 7 with parameters [ 63 , 60 , 2 ] . By Theorem 4.6, we obtain a quantum error-correcting code with parameters [ 63 , 57 , ≥ 2 ] 7 .

Example 4.4

Let n = 20 and R 2 = F 5 [ u 1 , u 2 ] / ⟨ u i 3 = u i , u i u j = u j u i ⟩ , ς 1 = ( 1 − u 1 2 ) ( 1 − u 2 2 ) , ς 2 = ( 1 − u 1 2 ) ( u 2 2 + u 2 ) 2 , ς 3 = ( 1 − u 1 2 ) ( u 2 2 − u 2 ) 2 , ς 4 = ( 1 − u 2 2 ) ( u 1 2 + u 1 ) 2 , ς 5 = ( u 1 2 + u 1 ) ( u 2 2 + u 2 ) 4 , ς 6 = ( u 1 2 + u 1 ) ( u 2 2 − u 2 ) 4 , ς 7 = ( 1 − u 2 2 ) ( u 1 2 − u 1 ) 2 , ς 8 = ( u 1 2 − u 1 ) ( u 2 2 + u 2 ) 4 , and ς 9 = ( u 1 2 − u 1 ) ( u 2 2 − u 2 ) 4 .

x 20 − 1 = ( x − 1 ) 4 ( x − 2 ) 5 ( x − 3 ) 5 ( x − 1 ) 5 , x 20 + 1 = ( x 2 − 3 ) 5 ( x 2 − 2 ) 5 .

Let C be a ( 1 − 2 u 1 2 − 2 u 2 2 − 2 u 1 2 u 2 2 ) -constacyclic code of length 20 over R 2 . g 1 = ( x − 3 ) 2 , g 2 = g 3 = ⋯ = g 9 = x 2 − 3 . By Theorem 4.5, we have C ⊥ ⊆ C , ϕ 2 ( C ) is a linear code over F 5 with parameters [ 180 , 162 , 3 ] . By Theorem 4.6, we obtain a quantum error-correcting code with parameters [ 180 , 144 , ≥ 3 ] 5 .

5 Conclusion

In this article, by studying the structure of constacyclic codes over R k = F q [ u 1 , u 2 , … , u k ] / ⟨ u i 3 = u i , u i u j = u j u i ⟩ , we construct some non-binary quantum codes from constacyclic codes over the finite non-chain ring R k .

Acknowledgements

The authors would like to thank the referees and the editor for their careful reading of the article and for valuable comments and suggestions, which improved the presentation of this manuscript.

Funding information: This work was supported by the Key Technologies Research and Development Program of Henan Province (No. 212102210573), the Key Scientific Research Project Plan of Henan Province colleges and universities (No. 20A110015), and Zhengzhou Special Fund for Basic Research and applied basic research (No. ZZSZX202111).
Author contributions: The authors applied the SDC approach for the sequence of authors.
Conflict of interest: The authors declare no conflict of interest.

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