Construction of quaternary quantum error-correcting codes via orthogonal arrays

This paper presents a method based on orthogonal arrays of constructing pure quaternary quantum error-correcting codes. As an application of the method, some infinite classes of quantum error-correcting codes with distances 2, 3, and 4 can be obtained. Moreover, the infinite class of quantum codes with distance 2 is optimal. The advantage of our method also lies in the fact that the quantum codes we obtain have less items for a basis quantum state than existing ones.

1 Introduction

Quantum systems are more fragile than classical systems. When quantum information travels across a noisy channel, errors are unavoidable [1–3]. The primary tool to deal with different types of quantum noises is quantum error-correcting codes (QECCs) [1, 2, 4, 5]. They play an important role in quantum information tasks, such as in entanglement purification, quantum key distribution, fault-tolerant quantum computation, and so on [6–8]. Since its discovery, code construction has come a long way [9–16]. Plenty of binary QECCs have been obtained, some of which from classical error-correcting codes (CECCs) [16–19]. Relatively speaking, there are still less studies on quaternary QECCs. We are motivated by the fact that CECCs are one-to-one connected to orthogonal arrays (OAs) [20]. It would be interesting to see if OAs can reciprocate and help QECCs, especially, quaternary ones. Therefore, the main aim of this work is to construct quaternary QECCs from OAs.

If L is an r × N array with elements from S = {0, 1, …, s − 1} and every r × k subarray of L contains each k-tuple based on S as a row with same frequency, then the array is said to be an orthogonal array of strength k (for some k in the range 0 ≤ k ≤ N). We will use OA (r, N, s, k) to denote such an array [21]. The theory of OAs has been developed significantly since the seminal work of Rao [22]. In particular, in recent years many new methods for constructing strength k OAs have been proposed, and a lot of new classes of OAs have been presented [23–29]. An OA (r, N, s, k) is said to be an irredundant orthogonal array (IrOA), if every row in any r × (N − k) subarray is unique [20]. If all of a pure quantum state’s reductions to k qudits are maximally mixed, it is said to be k-uniform. And this state consists of N subsystems with d levels. A connection between a k-uniform state and an irredundant orthogonal array (IrOA) was established by Goyeneche et al. [20]. For simplicity, the normalization factors are omitted from this paper.

Lemma 1. [20] If $L=\left(\begin{array}{cccc}\hfill s_1^1\hfill & \hfill s_2^1\hfill & \hfill \cdots \hfill & \hfill s_N^1\hfill \\ \hfill s_1^2\hfill & \hfill s_2^2\hfill & \hfill \cdots \hfill & \hfill s_N^2\hfill \\ \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill \cdots \hfill & \hfill ⋮\hfill \\ \hfill s_1^r\hfill & \hfill s_2^r\hfill & \hfill \cdots \hfill & \hfill s_N^r\hfill \end{array}\right)$ is an IrOA(r, N, s, k), then the superposition of r product states,

$|\mathrm{\Phi }\rangle =|s_1^1{s}_2^1\dots s_N^1\rangle +|s_1^2{s}_2^2\dots s_N^2\rangle +\cdots +|s_1^r{s}_2^r\dots s_N^r\rangle$

is a k-uniform state.By using this connection in Lemma 1, a lot of k-uniform states have been constructed from OAs [20, 30–37]. This kind of k-uniform states is closely related to QECCs [12, 20]. Usually, quantum information theory benefits from OAs [38–43]. These new developments in OAs and uniform states provide a higher possibility to construct infinite classes of QECCs from OAs [35, 36].In this work, we present a method based on OAs of constructing pure quaternary QECCs. As an application of the method, some infinite classes of QECCs with distances 2, 3, and 4 can be obtained. We know that quantum bound reflects the optimality of QECCs and is a key parameter to judge whether a construction method is effective or not. Moreover, the resulting infinite class of quantum codes with distance 2 is optimal. The advantage of our method also lies in the fact that the constructed QECCs have less items in each basis state than existing ones.This paper is organized as follows. After introducing symbols, definitions, and required lemmas in Section 2, Section 3 presents the main results. The conclusion is drawn in Section 4.

2 Preliminaries

We introduce several symbols, definitions and lemmas used in this paper.

Let ${\mathbb{F}}_4^n$ denote the n dimensional space over a Galois field ${\mathbb{F}}_4={\mathbb{F}}_2(\omega )=\{\mathrm{0,1},\omega ,\bar{\omega }={\omega }^2\}$, ω² = ω + 1. For the convenience of codeword expression, we use ${\mathbb{F}}_4=\{\mathrm{0,1,2,3}\}$. A^T is the transposition of matrix A. $(s)={(\mathrm{0,1},\dots ,s-1)}_{1\times s}^T$, and 0_r and 1_r represent the r × 1 vector of 0s and 1s, respectively. We define the Kronecker product A ⊗ B and the Kronecker sum A ⊕ B as $A\otimes B={(a_{ij}\cdot B)}_{mu\times nv}$ and $A\oplus B={(a_{ij}+B)}_{mu\times nv}$, respectively, if $A={(a_{ij})}_{m\times n}$ and $B={(b_{rs})}_{u\times v}$ with entries from a finite field with binary operations (+ and ⋅). Here, a_ij + B denotes the u × v matrix with elements a_ij + b_rs (1 ≤ r ≤ u, 1 ≤ s ≤ v). And if necessary, matrix A can always be viewed as a set of its row vectors. The strength of an orthogonal array L is denoted by t(L). We also use a k-strength OA to denote an OA of strength k for k ≥ 0. Let ${({\mathbb{C}}^4)}^{\otimes N}=\underset{N}{\underbrace{{\mathbb{C}}^4\otimes {\mathbb{C}}^4\otimes \cdots \otimes {\mathbb{C}}^4}}$.

Definition 1. [44] Suppose S^l = {(u₁, …, u_l)|u_i ∈ S, i = 1, 2, …, l}. The number of positions in which two vectors v = (v₁, …, v_l), u = (u₁, …, u_l) ∈ S^l differ from one another is defined as the Hamming distance HD(u, v) between them. Let HD(L) represents all possible values of the Hamming distance between two distinct rows of an OA L. The minimal distance of a matrix A means the minimal Hamming distance between its distinct rows and denoted by MD(A).Let k ≥ 1 and ${\mathcal{A}}^k$ represent the additive group of order s^k which consists all k-tuples of elements from $\mathcal{A}$. The typical vector addition is used as the binary operation. If ${\mathcal{A}}_0^k=\{(x_1,x_2,\dots ,x_k):x_1=\cdots =x_k\in \mathcal{A}\}$, ${\mathcal{A}}_0^k$ is a subgroup of ${\mathcal{A}}^k$ of order s, and ${\mathcal{A}}_i^k$, i = 1, …, s^k−1–1 will be used to denoted its cosets.

Definition 2. [44] Let D be an r × c matrix with elements from $\mathcal{A}$. For every r × k submatrix of D, its rows are seen as entries of ${\mathcal{A}}^k$. If in the submatrix each set ${\mathcal{A}}_i^k$, i = 0, 1, …, s^k−1–1, is represented equally frequently, then the D is said to be a difference scheme of strength k. We use D_k(r, c, s) to denote such a matrix. When k = 2, we denote D_k(r, c, s) by D(r, c, s).

Definition 3. [29] Let L be an OA(r, N, s, k). Suppose the rows of L can be partitioned into u submatrices {L₁, L₂, …, L_u} such that each L_i is an $OA(\frac{r}{u},N,s,k_1)$ with k₁ ≥ 0. Then the set {L₁, L₂, …, L_u} is called an orthogonal partition of strength k₁ of L. In particular, {L₁, L₂, …, L_u} is called a strength k₁ orthogonal partition of a space ${\mathbb{F}}_p^n$ if $L={\mathbb{F}}_p^n$ .

Definition 4. Let D = D_k(r, c, s). A set of difference schemes {D₁, D₂, …, D_u} is called a k₁-strength orthogonal partition of D, if D_i⋂D_j = ø for i ≠ j and ${\bigcup }_{i=1}^u{D}_i=D$.

Lemma 2. [44] If s≢2 (mod  4), and s ≤ k, then the difference scheme D_k(s^k−1, k + 1, s) exists.

Lemma 3. [34] If L = OA(s^k, N, s, k), then MD(L) = N − k + 1.

Lemma 4. [45] (1) Let D = D_k(r, c, s). Then D ⊕ (s) = OA(rs, c, s, k).(2) Let D = D_k (m, n, s) and L = OA (r, N, s, k) for k = 2, 3. Then D ⊕ L = OA (mr, nN, s, k).

Lemma 5. [36] (Expansive replacement method) Assume that L_A is a k-strength OA with s levels in factor 1 and that L_B is a k-strength OA with s rows. After building a one-to-one mapping between the levels of factor 1 in L_A and the rows of L_B, we may construct an OA of strength k by substituting each level of factor 1 in L_A with the matching row from L_B .

Lemma 6. [44] For a prime power s ≥ 2, an OA(s^k, s + 1, s, k) exists if s ≥ k − 1 ≥ 0.

Lemma 7. [12] If the reductions of all states in a subspace Q of ${({\mathbb{C}}^s)}^{\otimes N}$ to any given k parties are equal, then Q is an ((N,K,k + 1))_s QECC, and vice versa. Furthermore, if any state in Q is k-uniform, then Q is pure, and vice versa.We can also define a QECC ((N,K,k + 1))_s according to Lemma 7, where N denotes the code length, K is the dimension of the encoding state, k + 1 denotes the distance, and s denotes the levels number. For s = 2, it is simply ((N, K, k + 1)).

Lemma 8. [46] (quantum Singleton bound) If K > 1 in an ((N,K,k + 1))_s then K ≤ s^N−2k. Similarly, a pure ((N,1,k + 1))_s satisfies 2k ≤ N.

Definition 5. A QECC such that the equality in Lemma 8 holds is called optimal.

3 Construction of ((N,K,k + 1))₄ QECC

This section provides a construction method of quaternary quantum error-correcting codes (QECCs) from orthogonal arrays (OAs). In Theorem 1, we use Lemma 4 (2) to construct QECCs with distance 2. Theorems 2 and 3 produce QECCs with distances 3 and 4 from the OAs with orthogonal partitions. In Theorem 4, we study the existence of QECCs with any distance by using a special construction of OAs.

Theorem 1. For every N ≥ 2, there is a QECC ((N,K,2))₄ for each integer 1 ≤ K ≤ 4^N−2 where the ${((N,4^{N-2},2))}_4$ code is optimal.Proof. When N ≥ 5, a difference scheme D = D_N−1(4^N−2, N, 4) exists by Lemma 2. Let L = D ⊕ (4) = OA (4^N−1, N, 4, N − 1). By Lemma 3, MD(L) = N − (N − 1) + 1 = 2. Set $D=\left(\begin{array}{c}\hfill d_1\hfill \\ \hfill d_2\hfill \\ \hfill ⋮\hfill \\ \hfill d_{4^{N-2}}\hfill \end{array}\right)$. Let L_i = d_i ⊕ (4) = OA (4, N, 4, 1) for i = 1, 2, …, 4^N−2. Then t (L_i) = 1, MD (L_i) = N.From Lemma 1, $L_1,L_2,\dots ,L_{4^{N-2}}$ can generate 4^N−2 1-uniform states $|{\phi }_1\rangle ,|{\phi }_2\rangle ,\dots ,|{\phi }_{4^{N-2}}\rangle$. They can be used as a set of orthogonal basis to generate a subspace Q of ${({\mathbb{C}}^4)}^{\otimes N}$. Thus Q is an optimal ${((N,4^{N-2},2))}_4$ code by Lemma 7 and Definition 5.In addition, for any integer 1 ≤ K ≤ 4^N−2–1, if Q_K is the subspace spanned by |φ₁⟩, …, |φ_K⟩, then it is a ((N,K,2))₄ code.When 1 < N < 5, we can construct the following QECCs ${((N,4^{N-2},2))}_4$.When N = 2, an optimal ((2,1,2))₄ code can be generated with a basis |φ⟩ = |01⟩ + |12⟩ + |23⟩ + |30⟩.When N = 3, take $D(\mathrm{4,3,4})=\left(\begin{array}{ccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 2\hfill \\ \hfill 0\hfill & \hfill 2\hfill & \hfill 3\hfill \\ \hfill 0\hfill & \hfill 3\hfill & \hfill 1\hfill \end{array}\right)=\left(\begin{array}{c}\hfill d_1\hfill \\ \hfill d_2\hfill \\ \hfill d_3\hfill \\ \hfill d_4\hfill \end{array}\right)$. Let A_i = d_i ⊕ (4) and $A=\left(\begin{array}{c}\hfill A_1\hfill \\ \hfill A_2\hfill \\ \hfill A_3\hfill \\ \hfill A_4\hfill \end{array}\right)$. Obviously, A and A_i are OAs for i = 1, 2, 3, 4. From Lemma 3, MD(A) = 2, and by Lemma 7, an optimal ((3,4,2))₄ QECC can be obtained from A₁, …, A₄.When N = 4, take $D(\mathrm{4,2,4})=\left(\begin{array}{cc}\hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 2\hfill \\ \hfill 0\hfill & \hfill 3\hfill \end{array}\right)=\left(\begin{array}{c}\hfill d_1\hfill \\ \hfill d_2\hfill \\ \hfill d_3\hfill \\ \hfill d_4\hfill \end{array}\right)$. Then B_4(i−1)+j = d_i ⊕ ((4) (j − 1) ⊕ (4)) is an OA (4, 4, 4, 1) for i, j = 1, 2, 3, 4 and $B=\left(\begin{array}{c}\hfill B_1\hfill \\ \hfill B_2\hfill \\ \hfill ⋮\hfill \\ \hfill B_{16}\hfill \end{array}\right)=D(\mathrm{4,2,4})\oplus {\mathbb{F}}_4^2$ is an OA (64, 4, 4, 2). By simple calculation, we have MD(B) = 2. By Lemma 7, an optimal ((4,16,2))₄ QECC can be obtained from B₁, …, B₁₆.Remark. The quantum codes obtained by Theorem 1 have less items in a basis state than existing ones. For example, every basis states of the ((3,4,2))₄ code has four items. It has far less number of items for a basis state than the ((3,4,2))₄ in [13]. Compared with the codes [[N,N − 2,2]]₄ in [47] for N = 9 + 6m with 0 ≤ m ≤ 165, we have the codes for all N ≥ 2.

Theorem 2. Suppose L is an OA(r, N, 4, 2) with MD(L) ≥ 3. A QECC ((N,K,3))₄ exists, if there are vectors b₁, b₂, …, b_K in ${\mathbb{Z}}_4^N$ that fulfill HD(b_u, b_v) ≥ 3 and |HD(b_u, b_v) − HD(L)|≥ 3 for u ≠ v.Proof: Let $X=\left(\begin{array}{c}\hfill X_1\hfill \\ \hfill X_2\hfill \\ \hfill ⋮\hfill \\ \hfill X_K\hfill \end{array}\right)$, where X_u = 1_r ⊗ b_u + L for 1 ≤ u ≤ K. Both X and X_u are 2-strength OAs. Let x₁ = b_u + l₁, x₂ = b_v + l₂ ∈ X for l₁, l₂ ∈ L. Then we can compute the Hamming distance (HD) between x₁ and x₂ and the minimum distance (MD) of X.

(1) HD (x₁, x₂) = MD(L) ≥ 3, if u = v, l₁ ≠ l₂.

(2) HD (x₁, x₂) = HD (b_u, b_v) ≥ 3, if u ≠ v, l₁ = l₂.

(3) If u ≠ v and l₁ ≠ l₂, we have HD (x₁, x₂) ≥HD (b_u + l₂, x₂) − HD (b_u + l₂, x₁) or HD (x₁, x₂) ≥HD (b_u + l₂, x₁) − HD (b_u + l₂, x₂), hence HD (x₁, x₂) ≥|HD (b_u, b_v) − HD(L)|≥ 3.

Therefore, MD(X) ≥ 3. We can obtain K states from {X₁, X₂, …, X_K} and Lemma 1. Let Q be a subspace of ${({\mathbb{C}}^4)}^{\otimes N}$ with the K states to be an orthogonal basis. Thus Q is a QECC ((N,K,3))₄ by Lemma 7.

Theorem 3. There exists a QECC ${((3p,4^{p-n+1},3))}_4$ with $\frac{4^{n-1}+2}{3}\le p\le \frac{4^n-1}{3}$ for n ≥ 3 and with 3 ≤ p ≤ 5 for n = 2.Proof. Let {D₁, D₂, D₃, D₄} be orthogonal partition of the difference scheme D (16, 3, 4) = (0₁₆, (4) ⊕ 0₄, 0₄ ⊕ (4)) and $\{L_1,L_2,\dots ,L_{4^{p-n}}\}$ be an orthogonal partition of strength two of ${\mathbb{F}}_4^p$. Let Y_i denote the ith row of ${\mathbb{F}}_4^{p-n}$ with $\frac{4^{n-1}+2}{3}\le p\le \frac{4^n-1}{3}$ for n ≥ 3. Take

$M=\left(\begin{array}{c}\hfill D_1\oplus L_1\hfill \\ \hfill ⋮\hfill \\ \hfill D_1\oplus L_{4^{p-n}}\hfill \\ \hfill D_2\oplus L_1\hfill \\ \hfill ⋮\hfill \\ \hfill D_4\oplus L_{4^{p-n}}\hfill \end{array}\right)=\left(\begin{array}{c}\hfill M_1\hfill \\ \hfill ⋮\hfill \\ \hfill M_{4^{p-n}}\hfill \\ \hfill M_{4^{p-n}+1}\hfill \\ \hfill ⋮\hfill \\ \hfill M_{4^{p-n+1}}\hfill \end{array}\right),$

where L_i = $(a_1,\dots ,a_n,((a_{n+1},\dots ,a_p)+1_{4^n}\otimes Y_i))$ for i = 1, 2, …, 4^p−n and (a₁, a₂, …, a_p) is an OA (4ⁿ, p, 4, 2).Because D_j is a 2-strength difference scheme and L_i is a 2-strength OA, it follows from Lemma 4 that M_k = D_j⊕ L_i is a 2-strength OA for k = 1, 2, …, 4^p−n+1. Let m₁ = d₁ ⊕ l₁, m₂ = d₂ ⊕ l₂ ∈ M_k for d₁, d₂ ∈ D_j, l₁, l₂ ∈ L_i. Then we have

$HD\left(m_1,m_2\right)=\left\{\begin{array}{c}3\cdot HD\left(l_1,l_2\right),if{d}_1=d_2,\hfill \\ p\cdot HD\left(d_1,d_2\right),if{l}_1=l_2,\hfill \\ \left(3-HD\left(d_1,d_2\right)\right)\cdot HD\left(l_1,l_2\right)+\left(p-HD\left(l_1,l_2\right)\right)\cdot HD\left(d_1,d_2\right),if{d}_1\ne d_2,l_1\ne l_2.\hfill \end{array}\right.$

Therefore, MD (M_k) ≥ 3 and M_k is an IrOA for any k. Furthermore, M is an OA and has strength two because it is equal to $D(\mathrm{16,3,4})\oplus {\mathbb{F}}_4^p$ after row permutations. Similarly, we can obtain MD(M) ≥ 3. From Lemma 1, M₁, M₂, …, $M_{4^{p-n+1}}$ can generate 4^p−n+1 states. They can be used as a basis to form a subspace Q of ${({\mathbb{C}}^4)}^{\otimes \mathrm{3p}}$. From Lemma 7, Q is a QECC ${((3p,4^{p-n+1},3))}_4$.Similarly, when 3 ≤ p ≤ 5 and n = 2, we can construct a ${((3p,4^{p-1},3))}_4$ QECC.

Example 1. Let the following + be the operation in ${\mathbb{F}}_4$. Let $D_1=\left(\begin{array}{ccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 2\hfill \\ \hfill 0\hfill & \hfill 2\hfill & \hfill 3\hfill \\ \hfill 0\hfill & \hfill 3\hfill & \hfill 1\hfill \end{array}\right)$, $D_2=\left(\begin{array}{ccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 3\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 2\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 3\hfill & \hfill 2\hfill \end{array}\right)$, $D_3=\left(\begin{array}{ccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 2\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 2\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 3\hfill & \hfill 3\hfill \end{array}\right)$, $D_4=\left(\begin{array}{ccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 3\hfill \\ \hfill 0\hfill & \hfill 2\hfill & \hfill 2\hfill \\ \hfill 0\hfill & \hfill 3\hfill & \hfill 0\hfill \end{array}\right)$. In Theorem 3, we take 3 ≤ p ≤ 5 and n = 2. Let (a₁, a₂, …, a_p) be an OA (16, p, 4, 2) and L_i=(a₁, a₂, (a₃, …, a_p) + 1₁₆ ⊗ Y_i), where Y_i denotes the ith row of ${\mathbb{F}}_4^{p-2}$ for i = 1, 2, …, 4^p−2. Then $\{L_1,L_2,\dots ,L_{4^{p-2}}\}$ is an orthogonal partition of strength 2 of ${\mathbb{F}}_4^p$. We can obtain QECCs ${((9,4^2,3))}_4$, ${((12,4^3,3))}_4$ and ${((15,4^4,3))}_4$. With 6 ≤ p ≤ 21 for n = 3, Theorem 3 produces QECCs ${((18,4^4,3))}_4,{((21,4^5,3))}_4,\dots ,{((63,4^{19},3))}_4$.

Theorem 4. If an OA(4ⁿ, p, 4, 3) exists for p > n ≥ 3, then there is a ${((4p,4^{p-n+1},4))}_4$ QECC.Proof. This can be proved in the same way as Theorem 3.

Example 2. Let D₁ = (0₁₆, (4) ⊕ 0₄, 0₄ ⊕ (4), (4) ⊕ (4)), D₂ = (0₁₆, (4) ⊕ 0₄, 0₄ ⊕ (4), 1 + (4) ⊕ (4)), D₃ = (0₁₆, (4) ⊕ 0₄, 0₄ ⊕ (4), 2 + (4) ⊕ (4)), D₄ = (0₁₆, (4) ⊕ 0₄, 0₄ ⊕ (4), 3 + (4) ⊕ (4)). Then the difference scheme D₃(64, 4, 4) = (0₆₄, (4) ⊕ 0₁₆, 0₄ ⊕ (4) ⊕ 0₄, 0₁₆ ⊕ (4)) has a 3-strength orthogonal partition {D₁, D₂, D₃, D₄}. Take p = 5, 6 and n = 3 in Theorem 4. Let (a₁, a₂, …, a_p) be an OA(64, p, 4, 3) and L_i=(a₁, a₂, a₃, (a₄, …, a_p) + 1₆₄ ⊗ Y_i), where Y_i denotes the ith row of ${\mathbb{F}}_4^{p-3}$ for i = 1, 2, …, 4^p−3. Then $\{L_1,L_2,\dots ,L_{4^{p-3}}\}$ is an orthogonal partition strength 3 of ${\mathbb{F}}_4^p$. By Theorem 4, two new QECCs ${((20,4^3,4))}_4$ and ${((24,4^4,4))}_4$ can be obtained.

Theorem 5. Let L^N denote an OA(r, N, 4, k). Let $Y=[0_s\oplus L^{N_1},(s)\oplus L^{N_2}]$ for s ≤ 4 and N₁ + N₂ ≤ N. If MD(Y) ≥ k + 1, then there exists an ${((N_1+N_2,s,k+1))}_4$ QECC.Proof. Let $Y_i=[L^{N_1},i-1+L^{N_2}]$ for i = 1, 2, …, s. Since Y_i is isomorphic to Y₁, Y_i is an OA and t (Y_i) = k. And we have $Y=\left(\begin{array}{c}\hfill Y_1\hfill \\ \hfill Y_2\hfill \\ \hfill ⋮\hfill \\ \hfill Y_s\hfill \end{array}\right)$. If MD(Y) ≥ k + 1, then Y_i is an IrOA (r, N₁ + N₂, 4, k). By Lemma 7, an ${((N_1+N_2,s,k+1))}_s$ QECC exists.

Example 3. As illustrations for small size codes, we obtain ((6,2,3))₄, ((7,4,3))₄ and ((5,4,3))₄.Take an OA (32, 7, 4, 2) = (a₁, a₂, …, a₇) in [48]. For the case s = 2, take Y = (0₂ ⊕ (a₅, a₆), (2) ⊕ (a₂, a₃, a₄, a₇)). Then MD(Y) = 3. Application of Theorem 5 yields a new ((6,2,3))₄ code.Let s = 4 and Y = (0₄ ⊕ (a₄, a₅, a₆), (4) ⊕ (a₁, a₂, a₃, a₇)). Then MD(Y) = 3. By Theorem 5, we can construct a ((7,4,3))₄ code in [47].Let L⁵ = (a₁, a₂, …, a₅) be an OA (16, 5, 4, 2) and Y = (0₄ ⊕ (a₂, a₃), (4) ⊕ (a₁, a₄, a₅)). Then MD(Y) = 3 and we obtain an optimal ((5,4,3))₄ code from Theorem 5. Every basis states of the ((5,4,3))₄ code has 64 items. Compared to ((5,4,3))₄ in [14], it includes less items for its base states.

Theorem 6. Let L = OA(r, N, 4, k) with MD(L) ≥ k + 1. We can construct a QECC ((N,K,k + 1))₄ if there are vectors b₁, b₂, …, b_K in ${\mathbb{Z}}_4^N$ such that $\mathrm{\text{MD}}\left(\begin{array}{c}\hfill 1_r\otimes b_1+L\hfill \\ \hfill 1_r\otimes b_2+L\hfill \\ \hfill ⋮\hfill \\ \hfill 1_r\otimes b_K+L\hfill \end{array}\right)\ge k+1$.Proof: Let $M=\left(\begin{array}{c}\hfill M_1\hfill \\ \hfill M_2\hfill \\ \hfill ⋮\hfill \\ \hfill M_K\hfill \end{array}\right)=\left(\begin{array}{c}\hfill 1_r\otimes b_1+L\hfill \\ \hfill 1_r\otimes b_2+L\hfill \\ \hfill ⋮\hfill \\ \hfill 1_r\otimes b_K+L\hfill \end{array}\right)$. Evidently, MD(M) ≥ k + 1 and M_i is an OA (r, N, 4, k). By Lemma 7, there is a QECC ((N,K,k + 1))₄.

Example 4. For N = 7 and r = 32, take L = OA(32, 7, 4, 2) in [48]. We can get $b_1,b_2,\dots ,b_{10}\in {\mathbb{Z}}_4^7$ which meet the requirements in Theorem 6 where b₁ = (0000000), b₂ = (0001103), b₃ = (0011332), b₄ = (0012030), b₅ = (0013200), b₆ = (0020021), b₇ = (0022113), b₈ = (0023323), b₉ = (0030210), b₁₀ = (0031313). Then we can construct a new ((7,10,3))₄ QECC, which is better than the code ((7,4,3))₄ in [47].

Theorem 7. If m is an integer satisfying 4^m−1 + 3 < 2d ≤ 4^m + 3, then there exists a QECC ${((n_d,1,d))}_4$ for 2m(d − 1) ≤ n_d ≤ (4^m + 1)m.Proof: Let q = 4^m. From Lemma 6, there exists L_B = OA (q^d−1, q + 1, q, d − 1). By Lemma 3, MD (L_B) = q − d + 3. When the q levels, 0, 1, …, q − 1, are replaced respectively by distinct rows of ${\mathbb{Z}}_4^m$, we can construct L_C = OA (q^d−1 (q + 1)m, 4, d − 1). Removing the last 0, 1, 2, …, (q − 2d + 3)m columns from L_C, an L = OA (q^d−1, n_d, 4, d − 1) for 2m (d − 1) ≤ n_d ≤ (4^m + 1)m can be obtained and MD(L) ≥ d. By Lemma 7, the desired QECC ${((n_d,1,d))}_4$ exists.Remark. When m = 1, two optimal QECCs ((2,1,2))₄ and ((4,1,3))₄ can be obtained.

Example 5. By giving different values to d in Theorem 7, some new QECCs with larger distances can be obtained, which are listed in Table 1.

TABLE 1

TABLE 1. Some new QECCs with larger distance by Theorem 7.

Theorem 8. Construction of new codes ((16,1,6))₄, ((24,1,8))₄, ((23,81,5))₄, ((15,4,5))₄, ((14,16,4))₄, ((23,4,7))₄, ((20,256,4))₄ and ((6,1,4))₄ from Lemma 7.Proof: An IrOA (4⁸, 16, 4, 5) with MD = 6 obtained by using product of two OA (2⁸, 16, 2, 5)s in [49] and an IrOA (4¹², 24, 4, 7) with MD = 8 obtained by using product of two OA (2¹², 24, 2, 7)s in [49] can generate two new QECCs ((16,1,6))₄ and ((24,1,8))₄ respectively. By using product of two OA (4608,23,2,4)s obtained from the ((23,9,5)) QECC in Example 7 in [15], we can get an OA (4608², 23, 4, 4) with an orthogonal partition {C₁, C₂, …, C₈₁} of strength 4 which can generate a new QECC ((23,81,5))₄.An IrOA (4⁸, 15, 4, 5) with an orthogonal partition {A₁, A₂, A₃, A₄} of strength 4, an IrOA (4⁸, 14, 4, 5) with an orthogonal partition {B₁, B₂, …, B₁₆} of strength 3, an IrOA (4¹², 23, 4, 7) with an orthogonal partition {E₁, E₂, E₃, E₄} of strength 6 and an IrOA (4¹², 20, 4, 7) with an orthogonal partition {F₁, F₂, …, F₂₅₆} of strength 3 produce four new QECCs ((15,4,5))₄, ((14,16,4))₄, ((23,4,7))₄ and ((20,256,4))₄ respectively. In particular, an IrOA (64,6,4,3) in [48] yields an optimal QECC ((6,1,4))₄ in [50].

4 Conclusion

Binary QECCs have been widely studied, but the research on quaternary QECCs is still rare. In the study, from OAs we construct a large number of pure quaternary QECCs, some of which are optimal. The advantage of the method presented is that the quantum codes we obtain have fewer items for a basis quantum state compared with the existing ones. In future, we intend to construct more optimal QECCs with the distance $\ge 3$ and investigate the q-ary QECCs for other prime powers and non-primes q from OAs.

Data availability statement

The raw data supporting the conclusion of this article will be made available by the authors, without undue reservation.

Author contributions

Supervision, SP; conceptualization, SP and FY; investigation, SP, FY, RY, JD, and TW; methodology, FY and RY; validation, FY, RY, JD, and TW; writing—original draft, FY; writing—review and editing, SP and FY All authors have read and agreed to the published version of the manuscript.

Funding

This work is supported by National Natural Science Foundation of China (Grant Nos. 11971004, 622722 08, 62172196); Open Foundation of State Key Laboratory of Networking and Switching Technology, Beijing University of Posts and Telecommunications (Grant No. SKLNST-2022-1-01); Science and Tech-nology Research Project of Henan Province (202102210163); The Open Foundation of Guangxi Key Laboratory of Trusted Software (Grant No. KX202040).

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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