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

  • Bo Kong and Xiying Zheng EMAIL logo
Published/Copyright: September 12, 2022
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Open Mathematics
From the journal Open Mathematics Volume 20 Issue 1

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.

  1. 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).

  2. Author contributions: The authors applied the SDC approach for the sequence of authors.

  3. Conflict of interest: The authors declare no conflict of interest.

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Received: 2021-08-03
Revised: 2022-03-27
Accepted: 2022-06-24
Published Online: 2022-09-12

© 2022 Bo Kong and Xiying Zheng, published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/math-2022-0459
Keywords for this article
constacyclic codes; quantum codes; Gray map; dual-containing codes
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  58. Non-binary quantum codes from constacyclic codes over 𝔽q[u1, u2,…,uk]/⟨ui3 = ui, uiuj = ujui⟩
  59. On the general position number of two classes of graphs
  60. A posteriori regularization method for the two-dimensional inverse heat conduction problem
  61. Orbital stability and Zhukovskiǐ quasi-stability in impulsive dynamical systems
  62. Approximations related to the complete p-elliptic integrals
  63. A note on commutators of strongly singular Calderón-Zygmund operators
  64. Generalized Munn rings
  65. Double domination in maximal outerplanar graphs
  66. Existence and uniqueness of solutions to the norm minimum problem on digraphs
  67. On the p-integrable trajectories of the nonlinear control system described by the Urysohn-type integral equation
  68. Robust estimation for varying coefficient partially functional linear regression models based on exponential squared loss function
  69. Hessian equations of Krylov type on compact Hermitian manifolds
  70. Class fields generated by coordinates of elliptic curves
  71. The lattice of (2, 1)-congruences on a left restriction semigroup
  72. A numerical solution of problem for essentially loaded differential equations with an integro-multipoint condition
  73. On stochastic accelerated gradient with convergence rate
  74. Displacement structure of the DMP inverse
  75. Dependence of eigenvalues of Sturm-Liouville problems on time scales with eigenparameter-dependent boundary conditions
  76. Existence of positive solutions of discrete third-order three-point BVP with sign-changing Green's function
  77. Some new fixed point theorems for nonexpansive-type mappings in geodesic spaces
  78. Generalized 4-connectivity of hierarchical star networks
  79. Spectra and reticulation of semihoops
  80. Stein-Weiss inequality for local mixed radial-angular Morrey spaces
  81. Eigenvalues of transition weight matrix for a family of weighted networks
  82. A modified Tikhonov regularization for unknown source in space fractional diffusion equation
  83. Modular forms of half-integral weight on Γ0(4) with few nonvanishing coefficients modulo ℓ
  84. Some estimates for commutators of bilinear pseudo-differential operators
  85. Extension of isometries in real Hilbert spaces
  86. Existence of positive periodic solutions for first-order nonlinear differential equations with multiple time-varying delays
  87. B-Fredholm elements in primitive C*-algebras
  88. Unique solvability for an inverse problem of a nonlinear parabolic PDE with nonlocal integral overdetermination condition
  89. An algebraic semigroup method for discovering maximal frequent itemsets
  90. Class-preserving Coleman automorphisms of some classes of finite groups
  91. Exponential stability of traveling waves for a nonlocal dispersal SIR model with delay
  92. Existence and multiplicity of solutions for second-order Dirichlet problems with nonlinear impulses
  93. The transitivity of primary conjugacy in regular ω-semigroups
  94. Stability estimation of some Markov controlled processes
  95. On nonnil-coherent modules and nonnil-Noetherian modules
  96. N-Tuples of weighted noncommutative Orlicz space and some geometrical properties
  97. The dimension-free estimate for the truncated maximal operator
  98. A human error risk priority number calculation methodology using fuzzy and TOPSIS grey
  99. Compact mappings and s-mappings at subsets
  100. The structural properties of the Gompertz-two-parameter-Lindley distribution and associated inference
  101. A monotone iteration for a nonlinear Euler-Bernoulli beam equation with indefinite weight and Neumann boundary conditions
  102. Delta waves of the isentropic relativistic Euler system coupled with an advection equation for Chaplygin gas
  103. Multiplicity and minimality of periodic solutions to fourth-order super-quadratic difference systems
  104. On the reciprocal sum of the fourth power of Fibonacci numbers
  105. Averaging principle for two-time-scale stochastic differential equations with correlated noise
  106. Phragmén-Lindelöf alternative results and structural stability for Brinkman fluid in porous media in a semi-infinite cylinder
  107. Study on r-truncated degenerate Stirling numbers of the second kind
  108. On 7-valent symmetric graphs of order 2pq and 11-valent symmetric graphs of order 4pq
  109. Some new characterizations of finite p-nilpotent groups
  110. A Billingsley type theorem for Bowen topological entropy of nonautonomous dynamical systems
  111. F4 and PSp (8, ℂ)-Higgs pairs understood as fixed points of the moduli space of E6-Higgs bundles over a compact Riemann surface
  112. On modules related to McCoy modules
  113. On generalized extragradient implicit method for systems of variational inequalities with constraints of variational inclusion and fixed point problems
  114. Solvability for a nonlocal dispersal model governed by time and space integrals
  115. Finite groups whose maximal subgroups of even order are MSN-groups
  116. Symmetric results of a Hénon-type elliptic system with coupled linear part
  117. On the connection between Sp-almost periodic functions defined on time scales and ℝ
  118. On a class of Harada rings
  119. On regular subgroup functors of finite groups
  120. Fast iterative solutions of Riccati and Lyapunov equations
  121. Weak measure expansivity of C2 dynamics
  122. Admissible congruences on type B semigroups
  123. Generalized fractional Hermite-Hadamard type inclusions for co-ordinated convex interval-valued functions
  124. Inverse eigenvalue problems for rank one perturbations of the Sturm-Liouville operator
  125. Data transmission mechanism of vehicle networking based on fuzzy comprehensive evaluation
  126. Dual uniformities in function spaces over uniform continuity
  127. Review Article
  128. On Hahn-Banach theorem and some of its applications
  129. Rapid Communication
  130. Discussion of foundation of mathematics and quantum theory
  131. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part II)
  132. A study of minimax shrinkage estimators dominating the James-Stein estimator under the balanced loss function
  133. Representations by degenerate Daehee polynomials
  134. Multilevel MC method for weak approximation of stochastic differential equation with the exact coupling scheme
  135. Multiple periodic solutions for discrete boundary value problem involving the mean curvature operator
  136. Special Issue on Evolution Equations, Theory and Applications (Part II)
  137. Coupled measure of noncompactness and functional integral equations
  138. Existence results for neutral evolution equations with nonlocal conditions and delay via fractional operator
  139. Global weak solution of 3D-NSE with exponential damping
  140. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  141. Ground state solutions of nonlinear Schrödinger equations involving the fractional p-Laplacian and potential wells
  142. A class of p1(x, ⋅) & p2(x, ⋅)-fractional Kirchhoff-type problem with variable s(x, ⋅)-order and without the Ambrosetti-Rabinowitz condition in ℝN
  143. Jensen-type inequalities for m-convex functions
  144. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part III)
  145. The influence of the noise on the exact solutions of a Kuramoto-Sivashinsky equation
  146. Basic inequalities for statistical submanifolds in Golden-like statistical manifolds
  147. Global existence and blow up of the solution for nonlinear Klein-Gordon equation with variable coefficient nonlinear source term
  148. Hopf bifurcation and Turing instability in a diffusive predator-prey model with hunting cooperation
  149. Efficient fixed-point iteration for generalized nonexpansive mappings and its stability in Banach spaces
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