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Let [Formula: see text], where p is an odd prime and r is a positive integer. Consider a ring [Formula: see text], where [Formula: see text] with [Formula: see text] and [Formula: see text] with [Formula: see text]. In this paper, we examine the algebraic structure of constacyclic codes over the ring [Formula: see text] of block length [Formula: see text]. Further, a construction of quantum error-correcting codes (QECCs) from constacyclic codes over [Formula: see text] is also given. Moreover, we derive a number of new QECCs.

The main focus of this paper is to analyze the algebraic structure of constacyclic codes over the ring R = F p + w 1 F p + w 2 F p + w 2 2 F p + w 1 w 2 F p + w 1 w 2 2 F p , where w 1 2 − α 2 = 0 , w 1 w 2 = w 2 w 1 , w 2 3 − β 2 w 2 = 0 , and α , β ∈ F p ∖ { 0 } , for a prime p. We begin by introducing a Gray map defined over R , which is associated with an invertible matrix. We demonstrate its advantages over the canonical Gray map through some examples. Finally, we create new and improved quantum codes from constacyclic codes over R using Calderbank–Shore–Steane (CSS) construction. [ABSTRACT FROM AUTHOR]

This paper presents a study on the structure of 1-generator quasi-cyclic (QC) codes over the non-chain ring R = F q + u F q + v F q + u v F q , where u 2 = v 2 = 0 , u v = v u , and F q is a finite field of cardinality q = p r ; p is a prime. A minimal spanning set and size of these codes are determined. A sufficient condition for 1-generator QC codes over R to be free is given. BCH-type bounds on the minimum distance of free QC codes over R are also presented. Some optimal linear codes over F q are obtained as the Gray images of quasi-cyclic codes over R. Some characterizations of the Gray images of QC codes over R in F q and F q + u F q (u 2 = 0) are done. As an application, we consider self-orthogonal subcodes of the Gray images of QC codes over R to obtain new and better quantum codes than those are available in the literature. [ABSTRACT FROM AUTHOR]

A quaternionic Hadamard matrix (QHM) of order n is an n × n matrix H with non-zero entries in the quaternions such that H H ∗ = n I n , where I n and H ∗ denote the identity matrix and the conjugate-transpose of H, respectively. A QHM is dephased if all the entries in its first row and first column are 1, and it is non-commutative if its entries generate a non-commutative group. The aim of our work is to provide new constructions of infinitely many (non-commutative dephased) QHMs; such matrices are used by Farkas et al. (IEEE Trans Inform Theory 69(6):3814–3824, 2023) to produce mutually unbiased measurements. [ABSTRACT FROM AUTHOR]

Let q be a prime power and let ℛ = q [ u 1 , u 2 , ... , u k ] / 〈 f 1 (u 1) , ... , f k (u k) , u i u j − u j u i 〉 be a finite non-chain ring, where f i (u i) , 1 ≤ i ≤ k , are polynomials, not all linear, which split into distinct linear factors over q . We characterize constacyclic codes over the ring ℛ and study quantum codes from these. As an application, some new and better quantum codes, as compared to the best known codes, are obtained. We also prove that the choice of the polynomials f i (u i) , 1 ≤ i ≤ k , is irrelevant while constructing quantum codes from constacyclic codes over ℛ , it depends only on their degrees. [ABSTRACT FROM AUTHOR]

In this paper, we study skew constacyclic codes over a class of non-chain rings T = F q [ u 1 , u 2 , ... , u r ] / ⟨ f i (u i) , u i u j - u j u i ⟩ i , j = 1 r , where q = p m , p is some odd prime, m is a positive integer, and f i (u i) are non-constant, monic polynomials that split into distinct linear factors. We discuss the structural properties of skew constacyclic codes over T and their dual. We characterize Euclidean and Hermitian dual-containing skew constacyclic codes. These characterizations serve as a foundational framework for the development of techniques to construct quantum codes. Consequently, we derive plenty of new quantum codes including many maximum distance separable (MDS) quantum codes, and many quantum codes with better parameters than existing ones. Our work further extends to the characterization of skew constacyclic Euclidean and Hermitian linear complementary dual (LCD) codes over T , and we establish that their Gray images also preserve the LCD property. From this analysis, we derive numerous MDS codes and best known linear codes over F q . [ABSTRACT FROM AUTHOR]

In this paper, we provide a method to construct quantum codes from matrix-product codes which the constituent codes have no restrictions. Then, we construct several classes of quantum codes. Some of our quantum codes are new in the sense that the parameters are different from all the previous constructions. In addition, we have that the lengths of some quantum codes are same from the previous constructions, but these quantum codes have larger dimensions. [ABSTRACT FROM AUTHOR]

Let p be a prime number and q = p m for some positive integer m. In this paper, we find the possible Hermitian hull dimensions of λ -constacyclic codes over R e = F q 2 + u F q 2 + u 2 F q 2 + ⋯ + u e - 1 F q 2 , u e = 1 where F q 2 is the finite field of q 2 elements, e | (q + 1) and λ = η 1 α 1 + η 2 α 2 + ⋯ + η e α e for α l ∈ F q 2 ∗ of order r l such that r l ∣ q + 1 (for each 1 ≤ l ≤ e ). Further, we obtain some conditions for these codes to be Hermitian LCD. Also, under certain conditions, we establish a strong result that converts every constacyclic code to a Hermitian LCD code (Corollaries 2 and 3). We also study the structure of generator polynomials for Hermitian dual-containing constacyclic codes (Theorems 8 and 9), and obtain parameters of quantum codes using the Hermitian construction. The approach we used to derive Hermitian dual-containing conditions via the hull has not been used earlier. As an application, we obtain several optimal and near-to-optimal LCD codes, constacyclic codes having small hull dimensions, and quantum codes. [ABSTRACT FROM AUTHOR]

CSS-T codes were recently introduced as quantum error-correcting codes that respect a transversal gate. A CSS-T code depends on a CSS-T pair, which is a pair of binary codes (C 1 , C 2) such that C 1 contains C 2 , C 2 is even, and the shortening of the dual of C 1 with respect to the support of each codeword of C 2 is self-dual. In this paper, we give new conditions to guarantee that a pair of binary codes (C 1 , C 2) is a CSS-T pair. We define the poset of CSS-T pairs and determine the minimal and maximal elements of the poset. We provide a propagation rule for nondegenerate CSS-T codes. We apply some main results to Reed–Muller, cyclic and extended cyclic codes. We characterize CSS-T pairs of cyclic codes in terms of the defining cyclotomic cosets. We find cyclic and extended cyclic codes to obtain quantum codes with better parameters than those in the literature. [ABSTRACT FROM AUTHOR]

This paper presents some results on 1-generator quasi-cyclic (QC) codes and their duals over finite fields. We have explicitly obtained a set of generators for the dual of a 1-generator QC code C from the generator of C, for some restricted cases. From the form of the generators of the dual code C ⊥ , we have derived upper bounds on the minimum distance of C ⊥ . For their applications in constructing quantum codes, we have presented some criteria for 1-generator QC codes to be self-orthogonal or self-dual. Then using the CSS construction, we have obtained some new and better quantum codes with the help of self-orthogonal 1-generator QC codes. [ABSTRACT FROM AUTHOR]

In this paper, we propose a construction of quantum codes from Hermitian self-orthogonal matrix product codes over the finite fields. This construction is applied to obtain numerous new quantum codes, and all of them have higher rate than current quantum codes available. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in right (resp., left) quasi-polycyclic (QP) codes of length n = m ℓ with an associated vector a = (a 0 , a 1 , ... , a m - 1) ∈ F q m , which are a generalization of quasi-cyclic codes (QC) and quasi-twisted (QT) codes. They are defined as invariant subspaces of F q n by the right (resp., left) QP operator T ~ a → (resp., T ~ a ← ). A correspondence between the right (resp., left) ℓ -QP codes and the linear codes of length ℓ over the ring R a → , m : = F q [ x ] / x m - a → (x) , a → (x) = ∑ i = 0 m - 1 a i x i (resp., R a ← , m : = F q [ x ] / x m - a ← (x) , a ← (x) = ∑ i = 0 n - 1 a i x m - 1 - i ) is given. This correspondence leads to some basic characterizations of these codes such as generator and parity check polynomials. Moreover, we also discuss the structure of the 1-generator QP codes, and we prove BCH-like and the Hartmann–Tzeng-like bounds on the minimum distance of QP codes. Finally, we give examples of new quantum codes derived from QP codes as an application of some of the results. Several of these codes are MDS. [ABSTRACT FROM AUTHOR]

Let $ \mathfrak{R}_{l, k} = {\mathbb F}_{p^m}[u_1, u_2, \cdots, u_k]/ \langle u_{i}^{l} = u_{i}, u_iu_j = u_ju_i = 0 \rangle $, where $ p $ is a prime, $ l $ is a positive integer, $ (l-1)\mid(p-1) $ and $ 1\leq i, j\leq k $. First, we define a Gray map $ \phi_{l, k} $ from $ \mathfrak{R}_{l, k}^n $ to $ {\mathbb F}_{p^m}^{((l-1)k+1)n} $, and study its Gray image. Further, we study the algebraic structure of $ \sigma $-self-orthogonal and $ \sigma $-dual-containing constacyclic codes over $ \mathfrak{R}_{l, k} $, and give the necessary and sufficient conditions for $ \lambda $-constacyclic codes over $ \mathfrak{R}_{l, k} $ to satisfy $ \sigma $-self-orthogonal and $ \sigma $-dual-containing. Finally, we construct quantum codes from $ \sigma $-dual-containing constacyclic codes over $ \mathfrak{R}_{l, k} $ using the CSS construction or Hermitian construction and compare new codes our obtained better than the existing codes in some recent references.

In this paper, cyclic codes over F q 2 m of lengths 2 (q 2 m - 1) and q 2 m - 1 r (q - 1) are considered, where r ∣ (q + 1) and r ≠ q + 1 . Based on the Hermitian construction, the q 2 -ary images of these cyclic codes are employed to construct quantum codes. The quantum codes obtained by our technique have higher code rate than the currently known ones. [ABSTRACT FROM AUTHOR]

In this paper, let q be an odd prime power. Based on new constacyclic codes which contain their Hermitian duals and Hermitian construction, we construct some classes of quantum MDS codes and quantum codes. When q ≡ 1 mod 4 , x and y are a divisor of q - 1 and q + 1 , respectively, we can construct a class of new quantum codes of length n = 2 x y q 2 m - 1 q 2 - 1 for odd x , y , m ≥ 3 . These codes have larger dimensions than existing codes. In addition, for q with the form 2 a m ± (x 2 + y 2) a - 1 and odd x, y, a with g c d (x , y) = 1 , we get some quantum MDS codes of length n = q 2 + 1 a . [ABSTRACT FROM AUTHOR]

In the setting of entanglement-assisted quantum error-correcting codes (EAQECCs), the sender and the receiver have access to noiseless pre-shared entanglement. With this assumption, one may obtain codes with better information rates or improved error handling properties. Entanglement incurs costs and must be judiciously calibrated in designing quantum codes, relative to their deployment parameters. Revisiting known constructions, we devise tools from classical coding theory to better understand how the amount of entanglement can be varied. We present three new propagation rules for the parameters of these codes. Tables listing the parameters of qubit and qutrit EAQECCs that we can explicitly construct are supplied for reference and comparison. [ABSTRACT FROM AUTHOR]

In this article, Algebraic-Geometric (AG) codes and quantum codes associated with a family of curves that includes the famous Suzuki curve are investigated. The Weierstrass semigroup at some rational point is computed. Notably, each curve in the family turns out to be a Castle curve over some finite field and a weak Castle curve over its extensions. This is a relevant feature when codes constructed from the curve are considered. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of Hermitian dual-containing constacyclic codes of length n = q 2 m - 1 q + 1 , where q is a power of 2 and m ≥ 2 is an integer. We apply the Hermitian construction to gain quantum constacyclic codes, and lots of codes obtained in the present paper have parameters better than those of the known quantum codes. The Hermitian hull of a linear code is defined to be the intersection of itself and its Hermitian dual. We construct a family of entanglement-assisted quantum error-correcting (EAQEC) codes with n = 4 m - 1 3 via determining the dimensions of the Hermitian Hulls. [ABSTRACT FROM AUTHOR]

Current work presents a new approach to quantum surface codes on compact surfaces with genus g ≥ 2 using the identification of these surfaces with hyperbolic polygons and hyperbolic semi-regular tessellations. This method generalizes other contructions, and we show that this approach may give rise to codes with very good parameters. We present tables with several examples of these codes whose parameters had not been shown before. [ABSTRACT FROM AUTHOR]

In this paper, we introduce additive double polycyclic codes over F p 2 for any prime p. We study their algebraic structures and determine generator polynomials of these codes. We also give cardinality of them. Additionally, we establish that both Euclidean dual and Hermitian dual of an additive double polycyclic code over F p 2 are also an additive double polycyclic code over F p 2 . As practical applications of these codes, we exhibit additive double polycyclic codes over F 9 which contain triple as many codewords as optimal linear codes of same length and minimum distance over F 9 , and we derive optimal ternary linear codes and quantum codes from additive double polycyclic codes over F 9 . [ABSTRACT FROM AUTHOR]

The article titled "Quantum codes from ¿-dual-containing constacyclic codes over ¿¿,¿" explores the creation of quantum codes using constacyclic codes over a specific non-chain ring. The authors examine the properties of constacyclic codes and define a Gray map. They establish conditions for constacyclic codes to be self-orthogonal and dual-containing. The article then constructs quantum codes using the CSS or Hermitian construction and compares them to existing codes. This research contributes to the field of quantum information and coding theory. [Extracted from the article]

Let $ C $ be a $ (n,q^{2k},n-k+1)_{q^2} $ additive MDS code which is linear over $ {\mathbb F}_q $. We prove that if $ n \geq q+k $ and $ k+1 $ of the projections of $ C $ are linear over $ {\mathbb F}_{q^2} $ then $ C $ is linear over $ {\mathbb F}_{q^2} $. We use this geometrical theorem, other geometric arguments and some computations to classify all additive MDS codes over $ {\mathbb F}_q $ for $ q \in \{4,8,9\} $. We also classify the longest additive MDS codes over $ {\mathbb F}_{16} $ which are linear over $ {\mathbb F}_4 $. In these cases, the classifications not only verify the MDS conjecture for additive codes, but also confirm there are no additive non-linear MDS codes which perform as well as their linear counterparts. These results imply that the quantum MDS conjecture holds for $ q \in \{ 2,3\} $. [ABSTRACT FROM AUTHOR]

In this paper, we use the principal ideals of group rings to construct quantum codes. We use multivariate polynomial rings to represent group rings and use multivariate polynomials to represent codewords. We give a condition for multivariate polynomials such that they generate Hermitian dual-containing codes. We also find that the minimum distance of the code which is generated by a multivariate polynomial is related to the zeros distribution of the multivariate polynomial. After computer search and calculation, we get many quantum codes with good parameters over small finite fields. [ABSTRACT FROM AUTHOR]

In this paper we present a construction of quantum codes from 1-generator quasi-cyclic (QC) codes of index 2 over a finite field F q . We have studied QC codes of index 2 as a special case of F q -double cyclic codes. We have determined the structure of the duals of such QC codes and presented a necessary and sufficient condition for them to be self-orthogonal. A construction of 1-generator QC codes with good minimum distance is also presented. To obtain quantum codes from QC codes, we use the Calderbank-Shor-Steane (CSS) construction. Few examples have been given to demonstrate this construction. Also, we present two tables of quantum codes with good parameters obtained from QC codes over F q . [ABSTRACT FROM AUTHOR]

It is well-known that it is an important task to construct quantum codes with good parameters. In this paper, we first give the definition of the Euclidean sums of classical codes, and prove that the Euclidean sums of classical codes are Euclidean dual-containing. Then we construct five classes of quantum codes based on the Euclidean sums of the matrix-product codes. We compare our quantum codes with previously known quantum codes available in the literature. As can be seen, some of them have not been constructed before, and in some cases, have larger dimension than previous literature. [ABSTRACT FROM AUTHOR]

In the current paper, we study skew cyclic codes over the ring S : = F 2 × (F 2 + v F 2) with v 2 = v . We investigate the structural properties of skew cyclic codes over the ring S . Also, we describe the dual codes of skew cyclic codes with respect to the Euclidean inner product and the Hermitian inner product. Furthermore, we give a necessary and sufficient condition for a skew cyclic code over S to contain its dual. Using these results, we show that binary quantum error-correcting codes as well as entanglement-assisted quantum error-correcting codes can be constructed from skew cyclic codes over S . Finally, we give examples of binary entanglement-assisted quantum error-correcting codes with good parameters from skew cyclic codes over S . [ABSTRACT FROM AUTHOR]

Let q=pmq={p}^{m}, pp be an odd prime, and Rk=Fq[u1,u2,…,uk]/⟨ui3=ui,uiuj=ujui⟩{R}_{k}={{\mathbb{F}}}_{q}\left[{u}_{1},{u}_{2},\ldots ,{u}_{k}]\hspace{-0.08em}\text{/}\hspace{-0.08em}\langle {u}_{i}^{3}={u}_{i},{u}_{i}{u}_{j}={u}_{j}{u}_{i}\rangle , where k≥1k\ge 1 and 1≤i,j≤k1\le i,j\le k. In this article, we define a Gray map from Rkn{R}_{k}^{n} to Fq3kn{{\mathbb{F}}}_{q}^{{3}^{k}n}. We study constacyclic codes over Rk{R}_{k} and construct non-binary quantum codes over Fq{{\mathbb{F}}}_{q}.

In this work, we study cyclic codes of length n over a finite commutative non-chain ring ℛ = F q [ u , v ] / 〈 u 2 − γ u , v 2 − ϵ v , u v − v u 〉 where γ , ϵ ∈ F q * and we find new and better quantum error-correcting codes than previously known quantum error correcting codes. Then certain constraints are imposed on the generator polynomials of cyclic codes, so these codes become linear complementary dual codes (in short LCD codes). We then verify that the Gray image of linear complementary dual codes of length n over ℛ is a linear complementary dual code of length 4 n over F q by establishing a Gray map. [ABSTRACT FROM AUTHOR]

Let k , m be positive integers and F 2 m be a finite field of order 2 m of characteristic 2. The primary goal of this paper is to study the structural properties of cyclic codes over the ring S k = F 2 m [ v 1 , v 2 , ... , v k ] 〈 v i 2 − α i v i , v i v j − v j v i 〉 , for i , j = 1 , 2 , 3 , ... , k , where α i is the non-zero element of F 2 m . As an application, we obtain better quantum error correcting codes over the ring S 1 (for k = 1 ). Moreover, we acquire optimal linear codes with the help of the Gray image of cyclic codes. Finally, we present methods for reversible DNA codes. [ABSTRACT FROM AUTHOR]

This paper considers three kinds of quasi-cyclic codes of index two with one generator or two generators and their applications in quantum code construction. In accordance with the algebraic structure of linear codes, we determine the lower bounds of the symplectic weights of these quasi-cyclic codes. Quasi-cyclic codes with the dual-containing property enable the construction of quantum codes. Defining the coefficient symmetric polynomials of the generator polynomials gives a concise condition for the dual-containing of the quasi-cyclic codes. The lower bound results can significantly reduce the scope of the search for a larger minimum distance of quasi-cyclic codes. With these algebraic results and computer supports, we obtain classical quasi-cyclic codes with better parameters and some new quantum codes under the symplectic construction. In particular, two examples of the new quantum codes [ [ 63 , 42 , 6 ] ] 2 , [ [ 51 , 35 , 5 ] ] 2 improve the corresponding codes in Grassl's code table. [ABSTRACT FROM AUTHOR]

Let S k = F q [ u 1 , u 2 , ... , u k ] / 〈 u i 3 = u i , u i u j = u j u i = 0 〉 , where 1 ≤ i , j ≤ k , q = p m , p is an odd prime. First, we define two new Gray maps ϕ k and φ k , and study their Gray images. Further, we determine the structure of constacyclic codes and their dual codes, and give a necessary and sufficient conditions of constacyclic codes to contain their duals. Finally, we obtain some new quantum codes over F q by using CSS construction, and compare the constructed codes better than the existing literature. [ABSTRACT FROM AUTHOR]

In this paper, we study cyclic codes over the ring F p × (F p + v F p) , where p is an odd prime and v2 = v. We first investigate the properties of the ring F p × (F p + v F p) and the linear codes over this ring. We also define a distance-preserving Gray map from F p × (F p + v F p) to F p 3 . We discuss cyclic codes and their dual codes over the ring. Also, we define a set of generators for these codes. As an implementation, we show that quantum error-correcting codes can be obtained from dual containing cyclic codes over the ring by using the Calderbank-Shor-Steane (CSS) construction. Furthermore, we give some illustrative examples. Finally, we tabulate the non-binary quantum error-correcting codes obtained from cyclic codes over the ring. [ABSTRACT FROM AUTHOR]

In this paper, we study Hermitian dual-containing constacyclic codes over the finite ring R r = F q 2 + v 1 F q 2 + ⋯ + v r F q 2 , where q is a prime power and v i 2 = v i , v i v j = v j v i = 0 for 1 ≤ i,j ≤ r,i≠j. A necessary and sufficient condition is provided to determine whether a constacyclic code C over Rr is Hermitian dual-containing. Moreover, we propose a generalized Gray map ΦM to preserve the property of Hermitian dual-containing. Compared with some existing Gray maps, ΦM increases the possibility of making the minimum distance of ΦM(C) larger. As an application, some new quantum codes over F q are constructed from constacyclic codes over Rr. [ABSTRACT FROM AUTHOR]

We use symplectic self-dual additive codes over $ \mathbb{F}_4 $ obtained from metacirculant graphs to construct, for the first time, $ \left[\kern-0.15em\left[ {\ell, 0, d} \right]\kern-0.15em\right] $ qubit codes with parameters $ (\ell,d) \in \{(78, 20), (90, 21), (91, 22), (93,21),(96,22)\} $. Secondary constructions applied to the qubit codes result in many new qubit codes that perform better than the previous best-known. [ABSTRACT FROM AUTHOR]

Let p be a prime integer and m and s be two positive integers. We compute dual codes and Hamming distances of negacyclic codes of length p s over F p m + u F p m with u 2 = 0 . This yields new quantum codes from negacyclic codes, which are neither self-dual nor dual-containing codes. [ABSTRACT FROM AUTHOR]

Weight parity error correction (WPEC) is an error-correction scheme that can correct high-weight Pauli errors. Still, it was initially shown to apply only to Steane and Golay codes. In this work, we prove that the WPEC scheme is appropriate for concatenated codes constructed from two classes of quantum codes, quantum quadratic-residue codes and quantum stabilizer codes from self-orthogonal cyclic codes. We generalize the implementation of the WPEC technique to nonbinary quantum codes and provide an application using WPEC for the [ [ 121 , 1 , 25 ] ] 3 concatenated code. We also show that the WPEC may not be applicable for some quantum codes with specific parameters through an example of the [[15, 7]] code over F 2 . [ABSTRACT FROM AUTHOR]

Entanglement-assisted quantum-error-correcting (EAQEC) codes are quantum codes which use entanglement as a resource. These codes can provide better error correction than the (entanglement unassisted) codes derived from the traditional stabilizer formalism. In this paper, we provide a general method to construct EAQEC codes from cyclic codes. Afterwards, the method is applied to Reed–Solomon codes, BCH codes, and general cyclic codes. We use the Euclidean and Hermitian construction of EAQEC codes. Three families have been created: two families of EAQEC codes are maximal distance separable (MDS), and one is almost MDS or almost near MDS. The comparison of the codes in this paper is mostly based on the quantum Singleton bound. [ABSTRACT FROM AUTHOR]

Due to efficient encoding and decoding algorithms, BCH codes have many applications in data storage systems and satellite communications. Meanwhile, one advantage of BCH codes is to construct quantum codes. In this paper, we assume that q ≡ 1 mod m , where m is a positive integer. We first study a class of q - ary quantum codes of length r (q m - 1) q - 1 by CSS construction, where r ∣ q - 1 . Then, we construct two families of q - ary quantum codes of length r (q 2 m - 1) q 2 - 1 by Hermitian construction. These constructions produce new quantum codes with larger dimensions. [ABSTRACT FROM AUTHOR]

In this paper, we consider S-chains of extremal self-dual and self-orthogonal codes and their applications in the construction of quantum codes. Then, by virtue of covering radius, we determine necessary conditions for linear codes to have subcodes with large dual distances and design a new S-chain search method. As computational results, 18 S-chains with large distances are obtained, and many good quantum codes can be derived from those S-chains by Steane construction, some of which improve the previous results.

Let R r = F q + v 1 F F q + ··· + v r F q , where q is the power of prime, v i 2 = v i , v i v j = v j v i = 0 for 1 ≤ i, j ≤ r and r ≥ 1. In this paper, the structure of λ-constacyclic codes over the ring R r is studied and a Gray map ϕ from R r n to F q (r +1) n is given. The necessary and sufficient conditions for these codes to contain their Euclidean duals are determined. As an application, we obtain many new better quantum codes from dual-containing λ-constacyclic codes over R r , for r = 1, 2, 3, that improve the known existing quantum codes. [ABSTRACT FROM AUTHOR]

We introduce quantum pin codes: a class of quantum CSS codes. Quantum pin codes are a generalization of quantum color codes and Reed-Muller codes and share a lot of their structure and properties. Pin codes have gauge operators, an unfolding procedure and their stabilizers form so-called $\ell $ -orthogonal spaces meaning that the joint overlap between any $\ell $ stabilizer elements is always even. This last feature makes them interesting for devising magic-state distillation protocols, for instance by using puncturing techniques. We study examples of these codes and their properties. [ABSTRACT FROM AUTHOR]

Let q be a prime power and m ≥ 2 be a positive integer. Based on classical cyclic codes, we construct new quantum codes by using images of cyclic codes. Three classes of quantum codes are derived from the F q 2 images of cyclic codes over F q 2 m . Compared with the known ones, these new quantum codes have larger minimum distance or higher code rate. [ABSTRACT FROM AUTHOR]

One of the main objectives of quantum error-correction theory is to construct quantum codes with optimal parameters and properties. In this paper, we propose a class of 2-generator quasi-cyclic codes and study their applications in the construction of quantum codes over small fields. Firstly, some sufficient conditions for these 2-generator quasi-cyclic codes to be dual-containing concerning Hermitian inner product are determined. Then, we utilize these Hermitian dual-containing quasi-cyclic codes to produce quantum codes via the famous Hermitian construction. Moreover, we present a lower bound on the minimum distance of these quasi-cyclic codes, which is helpful to construct quantum codes with larger lengths and dimensions. As the computational results, many new quantum codes that exceed the quantum Gilbert-Varshamov bound are constructed over F q , where q is 2, 3, 4, 5. In particular, 16 binary quantum codes raise the lower bound on the minimum distance in Grassl’s table (Grassl in bounds on the minimum distance of linear codes and quantum codes. ). In nonbinary cases, many quantum codes are new or have better parameters than those in the literature. [ABSTRACT FROM AUTHOR]

In this paper, the structure of cyclic codes of odd length n over R = Z 4 + v Z 4 , v 2 = v , is conferred. We define a Gray map ξ , which is distance preserving from R n (Gray distance) to Z 2 4 n (Hamming distance), and show that it is Z 2 -linear. We construct quantum codes over Z 2 by utilizing the Gray images ξ (C) of cyclic codes C over R . As an application, we provide many new quantum MDS codes and quantum codes with good parameters comparing to known quantum codes. [ABSTRACT FROM AUTHOR]