Volume 2023, Issue 1 8196228

Research Article

Open Access

Quantum Codes Obtained from Skew ϱ-λ-Constacyclic Codes over

Bo Kong,

Bo Kong

School of Statistics and Mathematics, Henan Finance University, Zhengzhou 450046, China

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Xiying Zheng,

Corresponding Author

Xiying Zheng

[email protected]

orcid.org/0000-0003-4652-3555

Faculty of Engineering, Huanghe Science and Technology College, Zhengzhou 450063, China hhstu.edu.cn

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Jie Liu,

Jie Liu

Faculty of Engineering, Huanghe Science and Technology College, Zhengzhou 450063, China hhstu.edu.cn

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Hongjing Jiang,

Hongjing Jiang

Faculty of Engineering, Huanghe Science and Technology College, Zhengzhou 450063, China hhstu.edu.cn

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Bo Kong,

Bo Kong

School of Statistics and Mathematics, Henan Finance University, Zhengzhou 450046, China

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Xiying Zheng,

Corresponding Author

Xiying Zheng

[email protected]

orcid.org/0000-0003-4652-3555

Faculty of Engineering, Huanghe Science and Technology College, Zhengzhou 450063, China hhstu.edu.cn

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Jie Liu,

Jie Liu

Faculty of Engineering, Huanghe Science and Technology College, Zhengzhou 450063, China hhstu.edu.cn

Search for more papers by this author

Hongjing Jiang,

Hongjing Jiang

Faculty of Engineering, Huanghe Science and Technology College, Zhengzhou 450063, China hhstu.edu.cn

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First published: 22 March 2023

https://doi.org/10.1155/2023/8196228

Academic Editor: Liuguo Yin

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Abstract

Let , where q = p^m, p is an odd prime, α_i is a unit over , and i, j = 1,2, ⋯, k. In this article, we define a Gray map from to , we study the structure of skew ϱ-λ-constacyclic codes over , and then we give the necessary and sufficient conditions for skew ϱ-λ-constacyclic codes over to satisfy dual containing. Further, we have obtained some new nonbinary quantum codes from skew ϱ-λ-constacyclic over by using the CSS construction.

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 − ⋯−2u_m)-constacyclic codes over and Θ-λ-skew constacyclic codes over by applying the CSS construction [20, 21]. In [22, 23], some good quantum codes were obtained from linear skew constacyclic over and by using the Hermitian construction. In [24, 25], some new quantum codes were obtained from skew constacyclic codes over R_e,q[u]/〈u^e = 1〉, and some MDS quantum codes were construed from skew cyclic codes over 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 . In this article, we study the algebraic structures of skew ϱ-λ-constacyclic codes over ; as an application, we give some new quantum codes from skew ϱ-λ-constacyclic codes over by using the CSS construction.

The rest of this article is arranged as follows: In Section 2, we define a new nonchain ring and a Gray map from to and introduce some basic knowledge of skew constacyclic code over . In Section 3, we give the necessary and sufficient conditions for skew ϱ-λ-constacyclic codes over to satisfy dual containing. In Section 4, we give some examples and obtain some new quantum codes from skew ϱ-λ-constacyclic codes over .

2. Preliminaries

Let be a nonchain ring, where q = p^m, p is an odd prime and α_i is a unit over .

Clearly, is semilocal and has q^{(k + 1)} elements.

Let ς₁ = u₁/α₁, ς₂ = u₂/α₂, ⋯, ς_k = u_k/α_k, ς_k+1 = 1 − u₁/α₁ − u₂/α₂ − ⋯−u_k/α_k. We can get that ς_iς_j = 0, when i ≠ j, ς_iς_i = ς_i, when i, j = 1,2, ⋯, k + 1, and 1 = ς₁ + ς₂ + ⋯+ς_k+1. Thus, .

For any , r can only be said to r = r₁ς₁ + r₂ς₂ + ⋯+r_k+1ς_k+1, where and j = 1,2, ⋯, k + 1.

Let θ_t be

automorphism,

. We define the automorphism of

as follows:

()

By the above definition, the order of ϱ is m/t.

Let the set , the addition on is defined as the general form of polynomials and the multiplication of polynomials is (axⁱ)(bx^j) = aϱⁱ(b)x^i+j.

By the above definition, it is easy to know that the set is a noncommutative ring and a skew polynomial ring. g(x) is a right divisor of f(x) if there exists subject to f(x) = q(x)∗g(x). Similarly, the left divisor can be given as above.

Let λ be a unit of , the skew constacyclic shift σ_ϱ,λ of is defined by σ_ϱ,λ(c) = (λϱ(c_n−1), ϱ(c₀), ⋯, ϱ(c_n−2)). Then, C is called a skew ϱ-λ-constacyclic code of length n over if C is invariant under . In particular, C is called a skew ϱ-cyclic code and skew ϱ-negacyclic code of length n over when λ = 1 and λ = −1.

A map is defined as follows:

()

Then, is identified as a polynomial a₀ + a₁x + ⋯+a_n−1xⁿ⁻¹ over . Let the order of ϱ, |ϱ| = l; if l|n, we define a skew ϱ-λ-constacyclic code of length n over as a left ideal of .

By the same method of Lemma 7 in [17], we can have the following lemma.

Lemma 1. If , the centre of is and then .

For any

the Gray map ϕ_k is defined as follows:

()

We extend ϕ_k as follows:

()

where

For any c = (c₀, c₁, ⋯, c_n−1) ∈ C, c can be said to be as follows:

()

Let C be a linear code of length n over

, and

()

for j = 1,2, ⋯, k + 1. One can quickly verify that C_j is a linear code of length n over

for j = 1,2, ⋯, k + 1, and

Lemma 2 (see [14].)An element λ = λ₁ς₁ + λ₂ς₂ + ⋯+λ_k+1ς_k+1 is a unit in if and only if λ_j is a unit in for j = 1,2, ⋯, k + 1.

3. Skew ϱ-λ-Constacyclic Codes over

Lemma 3. Let be a linear code of length n over , and λ = λ₁ς₁ + λ₂ς₂ + ⋯+λ_k+1ς_k+1 is a unit in , ord(ϱ) | n. Then, ϱ(λ) = λ if and only if θ_t(λ_j) = λ_j and j = 1,2, ⋯, k + 1, where θ_t(α_i) = α_i and i = 1,2, ⋯, k.

Proof. Suppose ϱ(λ) = λ, we have

()

On comparing the coefficients, we have

()

Note that θ_t(α_i) = α_i for i = 1,2, ⋯, k, we can get that θ_t(λ_j) = λ_j for j = 1,2, ⋯, k + 1.

Conversely, if θ_t(λ_j) = λ_j for j = 1,2, ⋯, k + 1, note that θ_t(α_i) = α_i for i = 1,2, ⋯, k, then we can have θ_t(ς_i) = θ_t(u_i/α_i) = ς_i and θ_t(ς_k+1) = θ_t(1 − u₁/α₁ − ⋯−u_k/α_k) = ς_k+1.

So, ϱ(λ) = ϱ(λ₁ς₁ + λ₂ς₂ + ⋯+λ_k+1ς_k+1) = λ₁ς₁ + λ₂ς₂ + ⋯+λ_k+1ς_k+1 = λ.

Theorem 1. Let be a linear code of length n over and λ = λ₁ς₁ + λ₂ς₂ + ⋯+λ_k+1ς_k+1 is a unit in , ord(ϱ)|n,ϱ(λ) = λ. Then, C is a skew ϱ- λ-constacyclic code of length n over if and only if C_j is a skew θ_t- λ_j-constacyclic code of length n over for j = 1,2, ⋯, k + 1.

Proof. For any c_j = (c_j,0, c_j,1, ⋯, c_j,n−1) ∈ C_j, j = 1,2, ⋯, k + 1. Then,

If C_j is a θ_t- λ_j-constacyclic code of length n over , then

()

So, C is a skew ϱ-λ-constacyclic code of length n over .

On the other hand, if C is a skew ϱ-λ-constacyclic code of length n over , we have

()

So, , C_j is a skew θ_t- λ_j-constacyclic code of length n over for j = 1,2, ⋯, k + 1.

Theorem 2. Let be a skew ϱ- λ-constacyclic code of length n over λ = λ₁ς₁ + λ₂ς₂ + ⋯+λ_k+1ς_k+1 is a unit in ord(ϱ)|n,ϱ(λ) = λ. Then, is a skew θ_t- λ⁻¹-constacyclic code of length n over , and is a skew θ_t- -constacyclic code over for j = 1,2, ⋯, k + 1, where

Proof. Let be a skew ϱ- λ-constacyclic code of length n over , where λ = λ₁ς₁ + λ₂ς₂ + ⋯+λ_k+1ς_k+1 is a unit in . For any x = (x₀, x₁, ⋯, x_n−1) ∈ C^⊥, y = (y₀, y₁, ⋯, y_n−1) ∈ C, then

()

We can get that

()

; hence, C^⊥ is a skew θ_t- λ⁻¹-constacyclic code.

By Lemma 2, C^⊥ is a skew θ_t- λ⁻¹-constacyclic code of length n over . By Theorem 1, is a skew θ_t- -constacyclic code over for j = 1,2, ⋯, k + 1.

Theorem 3. Let be a skew ϱ- λ-constacyclic code of length n over λ = λ₁ς₁ + λ₂ς₂ + ⋯+λ_k+1ς_k+1 is a unit in ord(ϱ) | n,ϱ(λ) = λ. Then, there exists a polynomial subject to C = 〈ς₁g₁(x) + ς₂g₂(x) + ⋯+ς_k+1g_k+1(x)〉, where the right divisor of xⁿ − λ is ς₁g₁(x) + ς₂g₂(x) + ⋯+ς_k+1g_k+1(x), the generator polynomial of skew θ_t- λ_j-constacyclic C_j is , and g_j(x) divides xⁿ − λ_j on the right for j = 1,2, ⋯, k + 1.

Proof. Let be a skew ϱ-λ-constacyclic code of length n over By Theorem 1, C_j is a skew θ_t- λ_j-constacyclic code of length n over for j = 1,2, ⋯, k + 1.

Let g_j(x) be the generator polynomial of C_j, then

()

Let C^′ = 〈ς₁g₁(x) + ς₂g₂(x) + ⋯+ς_k+1g_k+1(x)〉. Clearly, C^′⊆C.

Because ς_j[(ς₁g₁(x) + ς₂g₂(x) + ⋯+ς_k+1g_k+1(x)] = ς_jg_j(x) for j = 1,2, ⋯, k + 1, so C⊆C^′.

Hence, C = C^′ = 〈ς₁g₁(x) + ς₂g₂(x) + ⋯+ς_k+1g_k+1(x)〉.

Because the right divisor of xⁿ − λ_j is g_j(x) for j = 1,2, ⋯, k + 1. Let f_j(x)g_j(x) = xⁿ − λ_j. Then, [ς₁f₁(x) + ς₂f₂(x) + ⋯+ς_k+1f_k+1(x)][ς₁g₁(x) + ς₂g₂(x) + ⋯+ς_k+1g_k+1(x)] = xⁿ − (λ₁ς₁ + λ₂ς₂ + ⋯+λ_k+1ς_k+1) = xⁿ − λ.

So, the right divisor of xⁿ − λ is ς₁g₁(x) + ς₂g₂(x) + ⋯+ς_k+1g_k+1(x).

Corollary 1. Let be a skew ϱ- λ-constacyclic code of length n over λ = λ₁ς₁ + λ₂ς₂ + ⋯+λ_k+1ς_k+1 is a unit in Then, , , where f_j(x)g_j(x) = xⁿ − λ_j, , , and is the generate polynomials of skew θ_t- -constacyclic for j = 1,2, ⋯, k + 1.

Proof. Let for j = 1,2, ⋯, k + 1, using Theorems 2 and 3, , then , and we can get that Let . Clearly, Because for j = 1,2, ⋯, k + 1, so .

Therefore,

4. Quantum Codes from Skew ϱ-λ-Constacyclic Codes over

Theorem 4. Let be a linear code of length n over with order |C| = q^l, and the minimum Gray distance of C is d_G. Then, ϕ_k(C) is a [(k + 1)n, l, d_G] linear code and ϕ_k(C)^⊥ = ϕ_k(C)^⊥. If C is a self-dual code over , then ϕ_k(C) is a self-dual code over .

Proof. By the definition of ϕ_k, we can have that ϕ_k(C) is a [(k + 1)n, l, d_G] linear code.

Let a = (a₀, a₁, ⋯, a_n−1) ∈ C, b = (b₀, b₁, ⋯, b_n−1) ∈ C^⊥,, j = 0,1,2, ⋯, n − 1, and a⁽ⁱ⁾ = (a_i,0, a_i,1, ⋯, a_i,n−1), b⁽ⁱ⁾ = (b_i,0, b_i,1, ⋯, b_i,n−1), i = 1,2, …, k + 1.Then, So and i = 1,2, ⋯, k + 1.Since ϕ_k(a) = (a⁽¹⁾, a⁽²⁾, ⋯, a^{(k + 1)})andϕ_k(b) = (b⁽¹⁾, b⁽²⁾, ⋯, b^{(k + 1)}),

()

So, we have ϕ_k(C^⊥)⊆ϕ_k(C)^⊥.

Because ϕ_k is bijective, |C| = |ϕ_k(C)|. Then, ∣ϕ_k(C^⊥) | = q^{(k + 1)n}/|C| = q^{(k + 1)n}/∣ϕ_k(C)∣ = ∣ϕ_k(C)^⊥∣. We have ϕ_k(C^⊥) = ϕ_k(C)^⊥.

If C is a self-dual code, C = C^⊥ and then ϕ_k(C)^⊥ = ϕ_k(C^⊥) = ϕ_k(C).

Therefore, ϕ_k(C) is a self-dual code over .

Lemma 4. Let C be a skew θ_t- λ-constacyclic code of length n over whose generator polynomial is g(x) and ord(θ_t)|n. Then, C contains its dual code if and only if xⁿ − λ is the right divisor of f^∗(x)f(x), where λ = ±1 and the generator polynomial of C^⊥ is f^∗(x).

Proof. Let C^⊥ = 〈f^∗(x)〉, where f(x)g(x) = (xⁿ − λ) and λ = ±1,C contains its dual code if and only if there exists subject to f^∗(x) = h(x)g(x), by Lemma 1, f^∗(x)f(x) = h(x)g(x)f(x) = h(x)f(x)g(x) = h(x)(xⁿ − λ) if and only if the right divisor of f^∗(x)f(x) is xⁿ − λ.

In the present section, we construct quantum codes from skew ϱ- λ-constacyclic over by using the CSS construction [1, 2].

Theorem 5. (CSS Construction). Let C = [n, k, d]q be a linear codes over , if C^⊥⊆C, then there exists a quantum code [[n, 2k − n, d]]_q.

Theorem 6. Let be a skew ϱ-λ-constacyclic code of length n over , ord(ϱ) | n, ϱ(λ) = λ, λ = λ₁ς₁ + λ₂ς₂ + ⋯+λ_k+1ς_k+1 is a unit in . Then, C^⊥⊆C if and only if the right divisor of is xⁿ − λ_i, λ_i = ±1 and i = 1,2, ⋯, k + 1.

Proof. Suppose the right divisor of is xⁿ − λ_i, by Lemma 4, , i = 1,2, ⋯, k + 1, then , which implies .

On the contrary, let C^⊥⊆C, then . Hence, . By Lemma 4, we have the right divisor of as xⁿ − λ_i, λ_i = ±1,i = 1,2, ⋯, k + 1.

Using Lemma 4 and Theorem 6, we can get the following corollary.

Corollary 2. Let be a skew ϱ- λ-constacyclic code of length n over , where ord(ϱ) | n, ϱ(λ) = λ, and λ = λ₁ς₁ + λ₂ς₂ + ⋯+λ_k+1ς_k+1 is a unit in Then, C^⊥⊆C if and only if j = 1,2, ⋯, k + 1.

Theorem 7. Let be a skew ϱ- λ-constacyclic code of length n over , where ord(ϱ) | n, ϱ(λ) = λ, and λ = λ₁ς₁ + λ₂ς₂ + ⋯+λ_k+1ς_k+1 is a unit in If C_j is a skew θ_t-λ_j-constacyclic code over and , where λ_i = ±1 and j = 1,2, ⋯, k + 1, then ϕ_k(C)^⊥⊆ϕ_k(C) and there exists a quantum code , where the minimum Gray weight of C is d_G and the dimension of ϕ_k(C) is l.

Proof. Since C_j is a skew θ_t-λ_j-constacyclic code over and , λ_j = ±1, j = 1,2, ⋯, k + 1, using Corollary 2, and C^⊥⊆C. So, ϕ_k(C^⊥)⊆ϕ_k(C), by Theorem 4ϕ_k(C)^⊥ = ϕ_k(C^⊥). Therefore, ϕ_k(C)^⊥⊆ϕ_k(C), and by Theorem 4, ϕ_k(C) = [(k + 1)n, l, d_G]. Using Theorem 5, there exists a quantum code .

Example 1. Let n = 3 and , , and is defined by θ_t(a) = a³, and Then, |ϱ| = 3, ord(ϱ)|n.

()

Let C be a skew ϱ- (ς₁ + ς₂ + (−1)ς₃)-constacyclic code of length 3 over . Let g(x) = ς₁g₁(x) + ς₂g₂(x) + ς₃g₃(x), where g₁(x) = g₂(x) = ω²x + ω², g₃(x) = x + ω. Then, C₁ = 〈g₁(x)〉 and C₂ = 〈g₂(x)〉 are skew negacyclic codes of length 3 over . C₃ = 〈g₃(x)〉 is a skew cyclic code of length 3 over . By Theorem 4, ϕ₂(C) = [9,6,3]₂₇. Using Theorem 7, C^⊥⊆C. So, we can get a quantum code [[9,3,3]]₂₇ such that n − k + 2 − 2d = 2.

Example 2. Let n = 8 and and is defined by θ_t(a) = a³, and Then, |ϱ| = 2, ord(ϱ)|n.

()

Let C be a skew ϱ- (ς₁ + ς₂ + (−1)ς₃ + (−1)ς₄)-constacyclic code of length 8 over . Let g(x) = ς₁g₁(x) + ς₂g₂(x) + ς₃g₃(x) + ς₄g₄(x), where g₁(x) = g₂(x) = ω²x³ + ω²x² + 2x + 1, g₃(x) = g₄(x) = ωx² + ω³x + 1. Then, C₁ = 〈g₁(x)〉 and C₂ = 〈g₂(x)〉 are skew cyclic codes of length 8 over . C₃ = 〈g₃(x)〉 and C₄ = 〈g₄(x)〉 are skew negacyclic codes of length 8 over . By Theorem 4, ϕ₃(C) = [32,22,4]₉. By Theorem 7, C^⊥⊆C. So, we can get a quantum code [[32,12,4]]₉.

Example 3. Let n = 12 and , and is defined by θ_t(a) = a³, and Then, |ϱ| = 2, ord(ϱ)|n.

()

Let C be a skew ϱ- (ς₁ + (−1)ς₂ + ς₃ + (−1)ς₄)-constacyclic code of length 12 over . Let g(x) = ς₁g₁(x) + ς₂g₂(x) + ς₃g₃(x) + ς₄g₄(x), where g₁(x) = g₃(x) = ω⁶x² + ω² andg₂(x) = g₄(x) = x² + ω². Then, C₁ = 〈g₁(x)〉 and C₃ = 〈g₃(x)〉 are skew cyclic codes of length 12 over , and C₂ = 〈g₂(x)〉 andC₄ = 〈g₄(x)〉 are skew negacyclic codes of length 12 over . By Theorem 4, ϕ₃(C) = [48,40,3]₉. By Theorem 7, C^⊥⊆C. So, we can get a quantum code [[48,32,3]]₉, which has larger dimension than [[48,24,3]]₉ in [21].

In Table 1, some new quantum codes are given from skew ϱ-λ constacyclic over . Our quantum codes [[18,12,3]]₂₇, [[9,3,3]]₂₇, [[18,12,3]]₄₇, [[18,12,3]]₁₆₉, [[28,22,3]]₄₉ 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 [[48,32,3]]₉, [[40,30,3]]₂₅, [[56,46,3]]₄₉ have larger dimensions than the quantum codes [[48,24,3]]₉, [[40,24,3]]₂₅, [[56,40,3]]₄₉ in [21].

Table 1. Quantum codes from skew ϱ-λ-constacyclic over

q	n	k	(λ₁, ⋯, λ_k+1)	<g₁(x), ⋯, g_k+1(x)>	ϕ_k(C)	[[n, l, d]]_q
49	6	2	(1, −1, −1)	(31, ω1, ω1)	[18,15,3]	[[18,12,3]]₄₉	n − k − 2d + 2 = 0
27	3	2	(1,1, −1)	(1ω, 1ω, ω²ω²)	[9,6,3]	[[9,3,3]]₂₇	n − k − 2d + 2 = 0
49	6	2	(1, −1, −1)	(31, ω1, ω1)	[18,15,3]	[[18,12,3]]₄₉	n − k − 2d + 2 = 0
169	6	2	(1, −1,1)	(ω¹⁶1, ω³⁸1, ω¹⁶1)	[18,15,3]	[[28,22,3]]₄₉	n − k − 2d + 2 = 0
49	14	1	(1, −1)	(1ω²²1, ω³1)	[28,25,3]	[[48,32,3]]₉	n − k − 2d + 2 = 0
9	12	3	(1, −1,1, −1)	(ω⁶0ω², 10ω², ω⁶0ω², 10ω²)	[48,40,3]	[[48,32,3]]₉	[[48,32,3]]₉ [21]
25	10	3	(1,1, −1, −1)	(ω⁸ω⁴1, ω⁴1, ω¹⁰1, ω¹⁰1)	[40,35,3]	[[40,30,3]]₂₅	[[40,24,3]]₂₅ [21]
49	14	3	(1, −1, −1,1)	(1ω²²1, ω³1, ω³1, ω⁶1)	[56,51,3]	[[56,46,3]]₄₉	[[56,46,3]]₄₉ [21]

5. Conclusions

In this article, we construct quantum codes by studying the structure of skew ϱ-λ-constacyclic codes over a finite nonchain ring , where q = p^m, p is an odd prime, and α_i is a unit over . The major contributions are as follows: we study the structure of skew ϱ-λ-constacyclic code of length n over 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 by using the CSS construction. Our obtained quantum codes are approached quantum MDS codes or have larger dimensions than [21].

Conflicts of Interest

The authors declare that there are no conflicts of interest.

Acknowledgments

This work was supported by the Zhengzhou Special Fund for Basic Research and Applied Basic Research (no. ZZSZX202111) and the Key Technologies Research and Development Program of Henan Province (no. 212102210573).

Open Research

Data Availability

All data generated or analysed during this study are included within the article.

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Quantum Codes Obtained from Skew ϱ-λ-Constacyclic Codes over

Abstract

1. Introduction

2. Preliminaries

3. Skew ϱ-λ-Constacyclic Codes over

4. Quantum Codes from Skew ϱ-λ-Constacyclic Codes over

5. Conclusions

Conflicts of Interest

Acknowledgments

Open Research

Data Availability

References

References

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Quantum Codes Obtained from Skew ϱ-λ-Constacyclic Codes over

Abstract

1. Introduction

2. Preliminaries

3. Skew ϱ-λ-Constacyclic Codes over

4. Quantum Codes from Skew ϱ-λ-Constacyclic Codes over

5. Conclusions

Conflicts of Interest

Acknowledgments

Open Research

Data Availability

References

References

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