New quantum codes from constacyclic codes over the ring $ R_{k,m} $

For any odd prime \begin{document}$ p $\end{document} , we study constacyclic codes of length \begin{document}$ n $\end{document} over the finite commutative non-chain ring \begin{document}$ R_{k,m} = \mathbb{F}_{p^m}[u_1,u_2,\dots,u_k]/\langle u^2_i-1,u_iu_j-u_ju_i\rangle_{i\neq j = 1,2,\dots,k} $\end{document} , where \begin{document}$ m,k\geq 1 $\end{document} are integers. We determine the necessary and sufficient condition for these codes to contain their Euclidean duals. As an application, from the dual containing constacyclic codes, several MDS, new and better quantum codes compare to the best known codes in the literature are obtained.


Introduction
Quantum computing is a fascinating topic for present research with a higher ability to solve severe problems faster than classical computers. The quantum errorcorrecting codes are used in the quantum computer to protect the quantum information from the noises that occurred during communication. After the pioneering work of Shor [35] in 1995, Calderbank et al. [5] proposed a prominent method to obtain quantum error-correcting codes from the classical error-correcting codes. The primary goal of this area is to construct better quantum codes employing state-of-art. In this connection, many significant works have been reported in the literature which provides better quantum codes over the finite fields, see [14,15,16,26,32]. It is also observed that the linear (e.g., cyclic, constacyclic) codes over finite non-chain rings produced a huge amount of good quantum codes [1,2,8,11,13,19,12,27,29,30]. In 2015, Ashraf and Mohammad [1] studied quantum codes from cyclic codes over F p + vF p . Meantime, Dertli et al. [8] presented some new binary quantum codes obtained from the cyclic codes over F 2 + uF 2 + vF 2 + uvF 2 , and then Ashraf and Mohammad [2] generalized their work over the ring F q + uF q + vF q + uvF q to derive new non-binary quantum codes. There are a lot of articles in which good quantum codes are obtained from the cyclic codes on different finite rings, see [11,13,19,23,31,33,32,34]. On the other side, recently, Gao and Wang [12], Li et al. [27], Ma et al. [29,30] considered the constacyclic codes over finite non-chain rings and obtained many new and better codes compare to the known codes. Based on the above studies, one can say that the constacyclic codes are a great resource to supply good quantum codes over finite rings. Hence, it is logical to study the constacyclic codes over new and different non-chain rings to construct more new quantum codes.
Towards this, we study the constacyclic codes over the family of commutative non-chain rings R k,m = F p m [u 1 , u 2 , . . . , u k ]/ u 2 i − 1, u i u j − u j u i i =j=1,2,...,k , where p is an odd prime and m, k are positive integers. Note that for k = 2, m = 1 the constacyclic codes over R k,m are studied in [21]. Further, authors constructed quantum codes based on cyclic codes over R 2,1 in [22] and over R 3,1 in [19], respectively. Therefore, the present article is a continuation and generalization of our previous studies in the context of new quantum codes construction. The main objective of the article is two-folded, first, we characterize the constacyclic codes over R k,m (Section 4), and then by utilizing the structures we obtain new and better quantum codes (Section 5). To do so, we define a new Gray map ψ which is different from the usual canonical map and capable to produce many quantum MDS codes (Table  1) and better quantum codes (Table [3][4][5] compare to the best-known codes in the literature.
The presentation of the article is organized as follows: In Section 2, the results related to finite rings along with some basic definitions and properties have been discussed. Section 3 gives the structure of constacyclic codes, while Section 4 presents the construction of quantum codes and many examples of better codes. Section 5 concludes the article.

Preliminary
Throughout the article, we use R k,m := F p m [u 1 , u 2 , . . . , u k ]/ u 2 i − 1, u i u j − u j u i where 1 ≤ i, j ≤ k, p is an odd prime and k, m are positive integers. Thus R k,m is a finite commutative ring (with unity) of characteristic p and order p 2 k m . Note that any element r ∈ R k,m has the expression r = α 0 + [7], R k,m has 2 k number of maximal ideals w 1 , w 2 , . . . , w k , where w i ∈ {1 − u i , 1 + u i }, 1 ≤ i ≤ k. Also, R k,m is a principal ideal ring where any ideal I = v 1 , v 2 , . . . , v t is principally generated by the element formed by the sum of all v i and their products, see ([7], Theorem 2.6). Therefore, by comparing above maximal ideals, we conclude that R k,m is a nonchain semi-local Frobenius ring. For instance, if k = 1, then there are two maximal ideals I 1 = 1 − u 1 , I 2 = 1 + u 1 in R 1,m and I 1 = I 2 . Clearly, R 1,m is a non-chain semi-local ring of order p 2m . On the other hand, R k,m contains (p m − 1) 2 k units which is discussed in Lemma 3.2.
Recall that a nonempty subset C of R n k,m is a linear code of length n over R k,m if it is an R k,m -submodule of R n k,m and each element of C is called a codeword. The rank of a code C over R k,m is the minimum number of elements which span C. If K is the rank, then the code C is said to be an [n, K] linear code. The Euclidean inner product of two elements a = (a 0 , a 1 , . . . , a n−1 ) and b = (b 0 , b 1 , . . . , b n−1 ) ∈ R n k,m is defined as a · b = n−1 i=0 a i b i . Let C be a linear code of length n over R k,m . Then the dual C ⊥ := {a ∈ R n k,m | a · b = 0 ∀ b ∈ C} is also a linear code. The code C is said to be self-orthogonal if C ⊆ C ⊥ and self-dual if C ⊥ = C.
Let A = {i 1 , i 2 , . . . , i s } be a subset of the set S = {1, 2, . . . , k} where i 1 < i 2 < · · · < i s and ς ∈ F p m such that 2 k ς ≡ 1 (mod p). We define where |A| = ∆ (1 ≤ ∆ ≤ k), and ∆ = 0 for A = φ. We use e 0 0 for e 0 φ = ς k ij =1 (1 + u ij ), which can be obtained from above. From the definition of e ∆ A , it is clear that the superscript ∆ is used to count the number of factors, like (1 − u ij ), present in e ∆ A . Let B be a subset of S different from A. Without loss of generality, let i j ∈ A and i j ∈ B. Then, from the construction of e ∆ A , we must say 1 − u ij divides e ∆ A , Again, by induction on k in R k,m , we have A⊆S e ∆ A = 1. In the light of the above discussion, we conclude that Therefore, {e A | A ⊆ S} is a set of pairwise orthogonal idempotent elements in R k,m . Hence, by Chinese Remainder Theorem, we decompose the ring R k,m as Thus, any element r ∈ R k,m , where can be uniquely written as is the set of all 2 k × 2 k invertible matrices over F p m . Now, we define a Gray map where, 1 ≤ i j ≤ k. Here, we enumerate the vector (β 0 , β i1 , β i2 , . . . , β i k , β i1,i2 , β i1,i3 , . . . , β i k −1,i k , . . . , β i1,i2...i k ) as (r 1 , r 2 , . . . , r 2 k ) = r. Then the map ψ is linear and can be extended from R n k,m to F 2 k n p m componentwise. The Hamming weight of a codeword c = (c 0 , c 1 , . . . , c n−1 ) ∈ C is denoted by wt H (c) and defined as the number of non-zero components in the codeword c. The Hamming distance for the code C is defined by d H (C) = min{d H (c, c ) | c = c , for all c, c ∈ C}, where d H (c, c ) is the Hamming distance between c, c ∈ C and d H (c, c ) = wt H (c − c ). Also, the Gray weight of any element r ∈ R k,m is define as wt G (r) = wt H (ψ(r)) and Gray weight forr = (r 0 , r 1 , . . . , r n−1 ) ∈ R n k,m is wt G (r) = n−1 i=0 wt G (r i ). Further, the Gray distance between codewords c, c ∈ C is defined as d G (c, c ) = wt G (c − c ).
It is worth mentioning that in earlier works [1,11,17,19], authors have used the canonical Gray maps which take every element into a vector consisting of its canonical components. But, we define the map ψ as the multiplication of a vector by an invertible matrix of order 2 k . Such type of Gray maps one can also find in [13,29,30] with respect to their setup. One of the main advantages to choose such Gray maps, like ψ, is to enhance the code parameters (particularly, dimension, and minimum distance, etc.) over the parameters obtained by the simple canonical Gray map. For example, using ψ we construct quantum code [ [22,2,7]] 5 in Example 4.6 whose minimum distance is larger than the quantum code [ [22,2,5]] 5 obtained in [17] under the usual canonical Gray map. Now, we present an example for k = 2 to understand the ring structure based on the set of pairwise orthogonal idempotent elements and Gray map discussed above.
Then R 2,m is a semi-local ring with four maximal ideals where 4ς ≡ 1 (mod p). Then we can write r uniquely as In this case, the Gray map ψ : Now, we review some important results on linear codes over R k,m . One can find the similar results in [7,19,30,36].
Theorem 2.2. The Gray map ψ defined in equation (1) is linear and weight preserving from R n k,m (Gray weight) to F 2 k n p m (Hamming weight).
Proof. Since the Gray map ψ is linear, ψ(C) is a linear code of length 2 k n. Also, the map ψ is distance preserving, hence ψ(C) is a [2 k n, K, d H ] linear code over the field F p m where d G = d H .
Hence, x · y = 0, and consequently ψ(C) is a self-orthogonal linear code of length 2 k n over F p m .
Let C be a linear code of length n over R k,m and for A ⊆ S, Then C A is a linear code of length n over F p m for all A ⊆ S. Also, C can be expressed as Moreover, the generator matrix for the code Proof. It follows the similar argument of ( [19], Theorem 5).

Constacyclic codes over R k,m
In this section, we discuss the structural properties of constacyclic codes over R k,m . These codes are used to obtain quantum codes in the subsequent section.
Conversely, let C A be a δ A -constacyclic code of length n over F p m , for A ⊆ S. Let r = (r 0 , r 1 , . . . , Hence, C is a γ-constacyclic code of length n over R k,m .
Proof. Let C = A⊆S e ∆ A C A be a γ-constacyclic code of length n over R k,m . Therefore, by Theorem 3.4, each C A is the δ A -constacyclic code of length n over F p m . Let Corollary 3.6. Every ideal of R k,m [x]/ x n − γ is principally generated.

Proof.
1. Let C = A⊆S e ∆ A C A be a γ-constacyclic code of length n over R k,m . Then, by Theorem 3.4, C A is a δ A -constacyclic code of length n over F p m , for all A ⊆ S. Therefore, C ⊥ A is a δ −1 A -constacyclic code over F p m . Hence,

New quantum codes and comparison
Recall that a q-ary quantum code of length n and size K is a K-dimensional subspace of q n -dimensional Hilbert space (C q ) ⊗n , where q = p m . Precisely, a quantum code is represented as [[n, k, d]] q , where n is the length, d is the minimum distance and K = q k . The quantum code [[n, k, d]] q satisfies the singleton bound k + 2d ≤ n + 2, and known as quantum MDS (maximum-distance-separable) if it attains the bound. In this section, we construct several new q-ary quantum codes by using the structure of γ-constacyclic codes over R k,m . Also, the necessary and sufficient conditions for these codes to contain their duals are obtained. We first recall the CSS construction (Lemma 4.1) which plays an important role to obtain the quantum codes.
where f * (x) is the reciprocal polynomial of f (x) and λ = ±1.
In the light of Lemma 4.1, one must say that the dual containing linear code is the key to obtain quantum codes under the CSS construction. Therefore, by using Lemma 4.2, we present the necessary and sufficient conditions of the constacyclic codes to contain their duals in the next result.
is a u 1 -constacyclic code of length 11 over R 1,1 . Let satisfying M M t = 2I 2 . Then the Gray image ψ(C) has the parameters [22,12,7]. Also, ( ), for i = 0, 1. Therefore, by Theorem 4.3, C ⊥ ⊆ C. Hence, by Theorem 4.5, there exists a quantum code [ [22,2,7]] 5 , which has the larger distance compare to the known code [ [22,2,5]] 5 given by [10,17]. Remark 1. In the above example, we have calculated that the Gray image ψ(C) is a [22,12,7] linear code over F 5 . Note that ψ(C) has length = 2 k n = 2 1 · 11 = 22 and dimension is equal to the sum of the dimensions of linear codes generated by polynomials f 0 (x) and f 1 (x)=6 + 6 = 12. Also, it has the generator matrix where M 0 , M 1 are generator matrices of linear codes generated by polynomials f 0 (x) and f 1 (x), respectively. Now, putting the generator matrix G of linear code ψ(C), we have computed the minimum distance 7 by the Magma computation system [4].
. Therefore, by Theorem 4.3, C ⊥ ⊆ C. Hence, by Theorem 4.5, there exists a quantum code [ [32,14,6]] 17 , which has larger dimension compare to the known code [ [32,12,6]] 17 given by [13]. Therefore, our code has larger code rate than the known. Then M satisfies M M t = 6I 4 and the Gray image ψ(C) has the parameters [16,12,4].  17 given in [30], we conclude that our code has larger distance than that code.
, we have x 6 − 1 =(x + 1)(x + 3)(x + 4)(x + 9)(x + 10)(x + 12) Then M satisfies M M t = 8I 4 and the Gray image ψ(C) has the parameters [24,16,6].  13 , which has larger distance than the known code [ [24,8,4]] 13 appeared in [13]. Table 2 gives the set of matrices over the finite field F p m , which are used to compute the Gray images of constacyclic codes in Table 1 and Table 3-5, respectively. Also, Table 1 presents some quantum MDS  codes while Table 3-5 include new and better quantum codes than previously known codes from the constacyclic codes over R 1,m = F p m [u 1 ]/ u 2 1 − 1 . In Table 1 and  Table 3-5, we used different columns as below: 1 st column-values of p m , 2 nd column-lengths of the codes, 3 rd column-values of the unit γ, 4 th column-corresponding values of the units δ 0 , δ 1 , 5 th &6 th column-generator polynomials for the constacyclic codes, 7 th column-used matrices to compute parameters of Gray images, 8 th column-parameters of Gray images ψ(C) of the constacyclic codes, 9 th column-parameters [[n, k, d]] p m of the obtained quantum codes, 10 th column-parameters [[n , k , d ]] p m of the best-known quantum codes.

Computation tables. The
In order to compare our obtained quantum codes with best-known codes, we include the 10 th column from different references as mentioned in the column. We have seen that our obtained codes given in the 9 th column are better than the codes shown in the 10 th column by means of larger code rates and larger minimum distances. To represent the generator polynomials f 0 (x), f 1 (x), we write their coefficients in decreasing order, e.g., we use 124114 to represent the polynomial x 5 + 2x 4 + 4x 3 + x 2 + x + 4.

Remark 2.
Recall that a code [[n, k, d]] q satisfying n − k + 2 − 2d = t is known as a quantum code with singleton defect t. Obviously t ≥ 0 and when t = 0, it is a quantum MDS code. Also, t is the judgmental parameter to determine a code how much close to MDS. In fact, smaller t implies code is close to MDS. Hence, the main objective should be to obtain the code whose t is closer to zero, as much as possible. In the above tables, we have seen that the quantum codes with singleton defect t = 2 are

Conclusion
In this article, we studied the constacyclic codes over the family of commutative non-chain rings R k,m to obtain new non-binary quantum codes over finite fields. In the above tables, we have determined many new quantum codes which are superior to the best-known codes in the literature. Therefore, we believe that our work will motivate researchers to unfold the existence of many new quantum codes which can be obtained from this class of constacyclic codes.