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Let λ = λ0 + uλ1 + ⋯ + uk−1λ k−1 be a Type 1 unit in ℜk = 픽pm + u픽pm + ⋯ + uk−1픽 pm(uk = 0), where p is an odd prime, m is a positive integer and λ0,λ1,…,λk−1 ∈ 픽pm,λ0≠0,λ1≠0. In this paper, we give the complete structure of all Type 1 λ-constacyclic codes and their duals of length 8ps over the finite commutative chain ring ℜk in terms of their generator polynomials. Using this structure, we determine the Hamming distance and the Rosenbloom–Tsfasman (RT) distance of all Type 1 λ-constacyclic codes. For pm ≡ 1(mod 4) and a unit λ ∈ ℜ2, we determine the b -symbol distances of all λ-constacyclic codes of length 8ps over ℜ2, where b ≤ 8. As illustrations, we provide several λ-constacyclic codes with new parameters with respect to Hamming, RT and b-symbol metrics. MDS codes are widely recognized for their optimal error-correction capability, and MDS b-symbol codes are generalization of MDS codes. We found some MDS b-symbol constacyclic codes of length 8ps over ℜ2. Additionally, for pm ≡ 1(mod 4), we provide a decoding algorithm for Type 1 constacyclic codes of length 8ps over ℜk with respect to the Hamming, RT and b-symbol metrics. [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]

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.

Let p be a prime such that pm ≡ 1(mod4), where m is a positive integer. For any nonzero element α of 픽pm, we determine the algebraic structure of all α-constacyclic codes of length 4ps over the finite commutative chain ring 픽pm[u] 〈u3〉 ≅픽pm + u픽pm + u2픽 pm, where u3 = 0 and s is a positive integer. If the unit α ∈ 픽pm is a square, α = δ2, each α-constacyclic code of length 4ps is expressed as a direct sum of an − δ-constacyclic code and an δ- constacyclic code of length 2ps. In the main case that the unit α is not a square, it is shown that any nonzero polynomial of degree at most 3 over 픽pm is invertible in the ambient ring (픽pm+u픽pm+u2픽pm)[x] 〈x4ps−α〉. It is also proven that the ambient ring (픽pm+u픽pm+u2픽pm)[x] 〈x4ps−α〉 is a local ring with the unique maximal ideal 〈x4 − α 0,u〉, where α0ps = α. Such α-constacyclic codes are then classified into eight distinct types of ideals, and the detailed structures of ideals in each type are provided. Among other results, the number of codewords, and the dual of each α-constacyclic code are obtained. The non-existence of self-dual and isodual α-constacyclic codes of length 4ps over 픽pm + u픽pm + u2픽 pm, when the unit α is not a square, is likewise proved. [ABSTRACT FROM AUTHOR]

Maximum distance separable (MDS) codes have the highest possible error-correcting capability among codes with the same length and size. Let γ be nonzero in F 2 m. We consider all cyclic and (1 + u γ) -constacyclic codes of length 2 s over F 2 m + u F 2 m with their Lee distance and investigate all the cases whether the corresponding Gray images are MDS by giving an analogue of the Singleton bound for codes over F 2 m + u F 2 m with the Lee distance through Gray map. [ABSTRACT FROM AUTHOR]

A code has locality r if a symbol in any coordinate of a codeword in the code can be recovered by accessing the value of at most r other coordinates. Such codes are called locally recoverable codes (LRCs for short). Since LRCs can recover a failed node by accessing the minimum number of the surviving nodes, these codes are used in distributed storage systems such as Microsoft Azure. In this paper, constacyclic LRCs are obtained from their parity-check polynomials. Constacyclic codes with locality r ≤ 2 and dimension 2 are obtained and a sufficient and necessary condition for these codes to have locality 1 is given. Then, the construction is generalized. Distance optimal constacyclic LRCs with distance 2 are obtained. Also, constacyclic codes with locality 1 are constructed. They may be so useful in practice thanks to their minimum locality. Constacyclic codes whose locality is equal to their dimension are given. Furthermore, constacyclic LRCs are obtained from cyclotomic cosets. [ABSTRACT FROM AUTHOR]

In this paper, δ -dual codes over finite commutative semi-simple rings are defined as a generalization of dual codes over finite commutative semi-simple rings. Some properties of δ -dual codes are given. We present necessary and sufficient conditions for a λ -constacyclic code of length n to be δ -self-dual, δ -self-orthogonal, δ -dual-containing, δ -LCD over finite commutative semi-simple rings. We also study the δ -dual of skew Θ - λ -constacyclic codes over finite commutative semi-simple rings. Among others, we also give necessary and sufficient conditions for a skew Θ - λ -constacyclic code of any length n to be δ -self-dual, δ -self-orthogonal, δ -dual-containing, δ -LCD over finite commutative semi-simple rings. [ABSTRACT FROM AUTHOR]

Let a and m be positive integers and λ be any unit in GR (p a , m) of the form λ = (σ 0 + p σ 1 + p 2 z) , where σ 0 , σ 1 ∈ T (p , m) \ { 0 } and z ∈ GR (p a , m) . The symbol-pair distance of all such λ -constacyclic codes over GR (p a , m) of length p s are determined. As an application, we identify all maximum distance separable (MDS) λ -constacyclic codes of length p s over GR (p a , m) with respect to the symbol-pair distance. We give numerous examples of newly constructed MDS symbol-pair codes, i.e., new optimal symbol-pair codes with respect to the Singleton bound. [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]

We study hulls of constacyclic codes of length n over a finite non-chain ring F q + v F q with respect to the Euclidean and Hermitian inner products, where v 2 = v . Under a special Gray map from F q + v F q to F q 2 , dimensions of hulls of Gary images of constacyclic codes are obtained. Some new quantum error-correcting codes (QECCs) with good parameters are constructed by the quantum construction X method under the Euclidean and Hermitian inner products, respectively. Some of these QECCs are MDS with the minimum distance greater than q 2 , and a few of these QECCs are MDS with the minimum distance equal to q. [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]

Due to the asymmetry of quantum errors, phase-shift errors are more likely to occur than qubit-flip errors. Consequently, there is a need to develop asymmetric quantum error-correcting (QEC) codes that can safeguard quantum information transmitted through asymmetric channels. Currently, a significant body of literature has investigated the construction of asymmetric QEC codes. However, the asymmetry of most QEC codes identified in the literature is limited by the dual-containing condition within the Calderbank-Shor-Steane (CSS) framework. This limitation restricts the exploration of their full potential in terms of asymmetry. In order to enhance the asymmetry of asymmetric QEC codes, we utilize entanglement-assisted technology and exploit the algebraic structure of cyclotomic cosets of constacyclic codes to achieve this goal. In this paper, we generalize the decomposition method of the defining set for constacyclic codes and apply it to count the number of pre-shared entangled states in order to construct four new classes of asymmetric entanglement-assisted quantum maximal-distance separable (EAQMDS) codes that satisfy the asymmetric entanglement-assisted quantum Singleton bound. Compared with the codes existing in the literature, the lengths of the constructed EAQMDS codes and the number of pre-shared entangled states are more general, and the codes constructed in this paper have greater asymmetry. [ABSTRACT FROM AUTHOR]

Maximum distance separable (MDS) codes are optimal in the sense that the minimum distance cannot be improved for a given length and code size. The most prominent MDS codes are generalized Reed–Solomon (GRS) codes. The square C 2 of a linear code C is the linear code spanned by the component-wise products of every pair of codewords in C . For an MDS code C , it is convenient to determine whether C is a GRS code by determining the dimension of C 2 . In this paper, we investigate under what conditions that MDS constacyclic codes are GRS. For this purpose, we first study the square of constacyclic codes. Then, we give a sufficient condition that a constacyclic code is GRS. In particular, we provide a necessary and sufficient condition that a constacyclic code of a prime length is GRS. [ABSTRACT FROM AUTHOR]

Let R = F p m + u F p m with u 2 = 0 , where m, s are positive integers and p is an odd prime. For any invertible element Λ of R , the symbol-pair distances of all Λ -constacyclic codes of length 2 p s over R are completely obtained. We identify all symbol-pair Maximum Distance Separable (MDS) constacyclic codes of length 2 p s over R . As examples, many new symbol-pair codes, as well as symbol-pair MDS codes are constructed. [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.

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]

The entanglement-assisted quantum error-correcting (EAQEC) codes have the potential to greatly generalize and enhance the performance of existing quantum error-correcting codes. In this paper, we investigate EAQEC codes of length q 2 - 1 r , where r is a positive divisor of q + 1 . Most of these codes are new, and some of them have better performance than ones obtained in the literature. The resulting EAQEC codes are maximum-distance-separable (MDS) if the minimum distance d ≤ n + 2 2 . [ABSTRACT FROM AUTHOR]

Let $ \mathbb{F}_q $ be the finite field with $ q = p^{k} $ elements, and $ p_{1}, p_{2} $ be two distinct prime numbers different from $ p $. In this paper, we first calculate all the $ q $-cyclotomic cosets modulo $ p_1p_2^t $ as a preparation for the following parts. Then we give the explicit generator polynomials of all the constacyclic codes of length $ p_1p_2^tp^s $ over $ \mathbb{F}_q $ and their dual codes. In the rest of this paper, we determine all self-dual cyclic codes of length $ p_1p_2^t p^s $ and their enumeration. This answers a question recently asked by B. Chen, H.Q.Dinh and Liu. In the last section, we calculate the case of length $ 5\ell p^{s} $ as an example.

Operator quantum error-correcting codes (OQECCs), also known as subsystem codes, can effectively protect quantum information from interference by encoding quantum information in the tensor factor of the subspace of the physical state space. Let R = F q 2 + v F q 2 with v 2 = v and q be an odd prime power. In this paper, by designing some flexible defining sets of the Gray images of Hermitian dual-containing constacyclic codes of length n = q 2 - 1 2 s over R, where s = h ℓ is a divisor of q + 1 , h ≥ 3 is an odd prime and ℓ is a positive integer, we construct two new infinite families of maximum distance separable (MDS) OQECCs with parameters: q 2 - 1 s , q 2 - 1 s - 2 d + 2 - r , r , d q , where 3 ≤ d ≤ q - 1 2 and 0 ≤ r < q 2 - 1 s - 2 d + 2 for ℓ even; q 2 - 1 s , q 2 - 1 s - 2 d + 2 - r , r , d q , where 3 ≤ d ≤ (q + 1) (s + 1) 2 s - 1 and 0 ≤ r < q 2 - 1 s - 2 d + 2 for ℓ odd. Notably, the parameters of these MDS OQECCs are new and not covered by well-known MDS OQECCs constructed from previous literature. [ABSTRACT FROM AUTHOR]

Entanglement-assisted quantum error-correcting codes (EAQECCs) can be constructed from any classical linear codes by using pre-existing entanglement between the sender and receiver. In this paper, we construct a new family of maximum distance separable (MDS) EAQECCs with flexible parameters via constacyclic codes over the finite non-chain ring F q 2 + u F q 2 + ⋯ + u r - 1 F q 2 , where q is a prime power and u r = 1 . Notably, our codes obtained are not covered by the codes available in the previous literature. [ABSTRACT FROM AUTHOR]

We recall a classic lower bound on the minimum Hamming distance of constacyclic codes over finite fields, analogous to the well-known BCH bound for cyclic codes. This BCH-like bound serves as a foundation for proposing some minimum-distance lower bounds for single-generator quasi-twisted (QT) codes. Associating each QT code with a constacyclic code over an extension field, we obtain the first bound. This is the QT analogue to a result in the literature for quasi-cyclic codes. We point out some weaknesses in this bound and propose a novel bound that takes into account the Chinese remainder theorem approach to QT codes as well as the BCH bound of constacyclic codes. This proposed bound, in contrast to previous bounds in the literature, does not presuppose a specific form of code generator and does not require calculations in any extension field. We illustrate that our bound meets the one in the literature when the code generator adheres to the specific form assumed in that study. Various numerical examples enable us to compare and discuss these bounds. [ABSTRACT FROM AUTHOR]

Let be the finite field with elements, and be two distinct prime numbers different from . In this paper, we first calculate all the -cyclotomic cosets modulo as a preparation for the following parts. Then we give the explicit generator polynomials of all the constacyclic codes of length over and their dual codes. In the rest of this paper, we determine all self-dual cyclic codes of length and their enumeration. This answers a question recently asked by B. Chen, H.Q.Dinh and Liu. In the last section, we calculate the case of length as an example. [ABSTRACT FROM AUTHOR]

Quantum convolutional codes, which are the correct generalization to quantum domain of their classical analogs, were introduced to overcome decoherence during long distance quantum communications. In this paper, we construct some classes of quantum convolutional codes via classical constacyclic codes. These codes are maximum-distance-separable (MDS) codes in the sense that they achieve the Singleton bound for pure convolutional stabilizer codes. Furthermore, compared with some of the codes available in the literature, our codes have better parameters and are more general. [ABSTRACT FROM AUTHOR]

We study the rank weight hierarchy, thus in particular the minimum rank distance, of polynomial codes over the finite field F q m , q a prime power, m ≥ 2 . We assume that polynomials involved are squarefree. We establish the rank weight hierarchy of [ n , n - 1 ] constacyclic codes. We characterize polynomial codes of rth rank weight r, and in particular of first rank or minimum rank distance 1. Finally, we provide a refinement of the Singleton bound, from which we show that cyclic codes cannot be MRD (maximum rank distance) codes, but constacyclic codes can be. [ABSTRACT FROM AUTHOR]

Let $p\not =3$ be any prime. In this paper, we completely determine symbol-triple distance of all $\gamma $ -constacyclic codes of length $3p^{s}$ over the finite commutative chain ring ${\mathcal{ R}}=\mathbb F_{p^{m}}+u\mathbb F_{p^{m}}$ , where $\gamma $ is a unit of $\cal R$ which is not a cube in $\mathbb {F}_{p^{m}}$ . We give the necessary and sufficient condition for a symbol-triple $\gamma $ -constacyclic code to be an MDS symbol-triple code. Using that, we establish all MDS symbol-triple $\gamma $ -constacyclic codes of length $3p^{s}$ over $\cal R$ . Some examples of the symbol-triple distance of $\gamma $ -constacyclic codes of length $3p^{s}$ over $\cal R$ are provided. We also list some new MDS symbol-triple $\gamma $ -constacyclic codes of length $3p^{s}$ over $\cal R$ , where $\gamma $ is not a cube in $\mathbb {F}_{p^{m}}$ .

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}.

Let R be a finite commutative chain ring with unity and λ be a unit in R. In this paper, all non-trivial linear complementary pair (LCP) of λ -constacyclic codes of arbitrary length over R have been completely determined. An expression for the total number of non-trivial LCP of λ -constacyclic codes of length n over R has also been derived in terms of the maximum number of factors of x n - λ into monic, pairwise coprime polynomials of degree ≥ 1 over R. Further, using the algebraic structure of λ -constacyclic codes over finite chain rings of nilpotency index 2 as an alternative approach, the complete characterization of non-trivial LCP of λ -constacyclic codes is obtained for such rings. As an illustration of our results, a few examples of non-trivial LCP of constacyclic codes over the rings Z 8 , Z 4 and the Galois ring GR(4, 3) have been given. [ABSTRACT FROM AUTHOR]

Let R be the finite chain ring R = F p m + u F p m (u 2 = 0) , where p is an odd prime number and m is a positive integer. For η ∈ F p m ∗ , the Hamming distances of all η -constacyclic codes of length 2 p s over R had already been studied in Dinh et al. (in AAECC, 2020. https://doi.org/10.1007/s00200-020-00432-0). However, such a study is incomplete. In this paper, we provide corrections to some results that appeared in Dinh et al. (2020) and we completely solve the problem of determination of the Hamming distance of η -constacyclic codes of length 2 p s over R . [ABSTRACT FROM AUTHOR]

Let p ≠ 3 be any prime. In this paper, we compute symbol-pair distance of all γ -constacyclic codes of length 3 p s over the finite commutative chain ring R = F p m + u F p m , where γ is a unit of R which is not a cube in F p m . We give the necessary and sufficient condition for a symbol-pair γ -constacyclic code to be an MDS symbol-pair code. Using that, we provide all MDS symbol-pair γ -constacyclic codes of length 3 p s over R. Some examples of the symbol-pair distance of γ -constacyclic codes of length 3 p s over R are provided. [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]

Entanglement-assisted quantum error-correcting (EAQEC) codes are a subclass of quantum error correcting codes which use entanglement as a resource. These codes can provide error correction capability higher than the codes derived from the traditional stabilizer formalism. It is possible to construct an EAQEC code from any classical linear code, unlike standard quantum error-correcting codes, which can only be constructed from dual-containing codes. However, the number c of pre-shared maximally entangled states is usually calculated by computer search. In this paper, we first give a new formula of the intersection dimension of two distinct constacyclic codes. Then, using this formula, we obtain a method of computing the number c of pre-shared maximally entangled states. Finally, to enrich the variety of available EAQEC codes, many new EAQEC codes, are constructed to illustrate our results. [ABSTRACT FROM AUTHOR]

Polyadic constacyclic codes over finite fields have been of interest due to their nice algebraic structures, good parameters, and wide applications. Recently, the study of Type-I m-adic constacyclic codes over finite fields has been established. In this paper, a family of Type-II m-adic constacyclic codes is investigated. The existence of such codes is determined using the length of orbits in a suitable group action. A necessary condition and a sufficient condition for a positive integer s to be a multiplier of a Type-II m-adic constacyclic code are determined. Subsequently, for a given positive integer m, a necessary condition and a sufficient condition for the existence of Type-II m-adic constacyclic codes are derived. In many cases, these conditions become both necessary and sufficient. For the other cases, determining necessary and sufficient conditions is equivalent to the discrete logarithm problem which is considered to be computationally intractable. Some special cases are investigated together with examples of Type-II polyadic constacyclic codes with good parameters. [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]

This paper presents a formula for the dimension of Galois hulls of constacyclic codes. For this, we have arranged the irreducible factors of xn − λ over the finite field F q in a suitable way. Also, considering some restrictions on q, the number of constacyclic codes of length n over F q is calculated for a given Galois hull dimension. [ABSTRACT FROM AUTHOR]

MDS symbol-pair codes forms an optimal class of symbol-pair codes for their best error-correction capability. ψ -constacyclic codes of length p s over R = F q + u F q + u 2 F q (u 3 = 0) , where q = p m , ψ is a nonzero element of F q , are precisely the ideals of the ring R [ x ] / ⟨ x p s - ψ ⟩ which is a local finite non chain ring with the non principal maximal ideal ⟨ u , x - φ ⟩ , where φ ∈ F q satisfying ψ = φ p s . In this paper, all MDS symbol-pair ψ -constacyclic codes of length p s over R are established. [ABSTRACT FROM AUTHOR]

Due to the asymmetry of quantum errors, phase-shift errors are more likely to occur than qubit-flip errors. Consequently, there is a need to develop asymmetric quantum error-correcting (QEC) codes that can safeguard quantum information transmitted through asymmetric channels. Currently, a significant body of literature has investigated the construction of asymmetric QEC codes. However, the asymmetry of most QEC codes identified in the literature is limited by the dual-containing condition within the Calderbank-Shor-Steane (CSS) framework. This limitation restricts the exploration of their full potential in terms of asymmetry. In order to enhance the asymmetry of asymmetric QEC codes, we utilize entanglement-assisted technology and exploit the algebraic structure of cyclotomic cosets of constacyclic codes to achieve this goal. In this paper, we generalize the decomposition method of the defining set for constacyclic codes and apply it to count the number of pre-shared entangled states in order to construct four new classes of asymmetric entanglement-assisted quantum maximal-distance separable (EAQMDS) codes that satisfy the asymmetric entanglement-assisted quantum Singleton bound. Compared with the codes existing in the literature, the lengths of the constructed EAQMDS codes and the number of pre-shared entangled states are more general, and the codes constructed in this paper have greater asymmetry.

Let γ = 4 z - 1 be an unit of Type (∗ -) of the Galois ring GR (2 a , m) . The γ -constacyclic codes of length 2 s over the Galois ring GR (2 a , m) are precisely the ideals ⟨ (x + 1) i ⟩ , 0 ≤ i ≤ 2 s a of the chain ring R (a , m , γ) = GR (2 a , m) [ x ] ⟨ x 2 s - γ ⟩ . This structure is used to determine the symbol pair distance of γ -constacyclic codes of length 2 s over GR (2 a , m) . The exact symbol-pair distances for all such γ -constacyclic codes of length 2 s over GR (2 a , m) are obtained. Also, we provide the MDS symbol-pair codes of length 2 s over GR (2 a , m) and some examples are computed. [ABSTRACT FROM AUTHOR]

In this paper, we study constacyclic codes over the ring R k = F q [ u 1 , u 2 , ⋯ , u k ] / ⟨ u i 3 - u i , u i u j - u j u i ⟩ , for all i , j = 1 , ... , k , q = p m for any odd prime p and positive integers m, k. We study the structure of the ring R k and define a Gray map by a matrix and decompose constacyclic codes over the ring R k as a direct sum of constacyclic codes over F q . We give the relevant properties of constacyclic codes over the ring R k and give necessary and sufficient conditions for constacyclic codes to be dual-containing. Taking R 2 as an example, we give some corresponding conclusions. As an application, we construct some new quantum error-correcting codes over F q from constacyclic codes over R k . [ABSTRACT FROM AUTHOR]

Classically, negacyclic codes over finite fields enjoy a good error-correcting capability with respect to the Lee metric. Lee distance of negacyclic codes of length 2 s over the finite commutative chain ring Z 2 e have been determined. Further, these results have been generalized to the class of (4 z - 1) -constacyclic codes of length 2 s over GR 2 e , m , z ∈ GR 2 e , m . In this paper, we obtain the Lee distance of a more general class of 1 + 2 σ 1 + 4 z -constacyclic codes of length 2 s over GR 2 e , m , where σ 1 ∈ T 2 , m \ { 0 } and z ∈ GR 2 e , m . As an application, we compute all maximum distance separable (MDS) codes with respect to the Lee distance. [ABSTRACT FROM AUTHOR]

Let q be an odd prime power, and w be a generator of the multiplicative cyclic group F q 2 ∗ . In this work, we study constacyclic codes over R = F q 2 + u F q 2 (u 2 = w 2) , and show a construction of quantum codes by studying the Hermitian construction over R. To do that, we discuss the σ -inner product over any finite commutative Frobenius ring and using that we define the Hermitian inner product over R. We decompose the ring R by constructing a pairwise orthogonal idempotent and study linear codes. We present the units of R and using them we study constacyclic codes and their generators over R. We also provide a condition for a constacyclic code to contain its Hermitian dual over this ring. Finally, we constructed some codes, which have improved parameters than some of the recently constructed codes in the literature. We also present a table of codes, which have better parameters than those currently available in the online database. [ABSTRACT FROM AUTHOR]

Let F q be a finite field of order q, t be a prime and m 1 , m 2 , m 3 be positive integers. In this article, we find all irreducible divisors of x l 1 m 1 l 2 m 2 l 3 m 3 − a over F q where a ∈ F q * and q t − 1 = l 1 v 1 l 2 v 2 l 3 v 3 c such that l 1 , l 2 , l 3 are distinct odd primes and c is a positive integer with gcd (l 1 l 2 l 3 , c) = 1 and gcd (l 1 l 2 l 3 , q (q − 1)) = 1. Moreover, we construct an a-constacyclic code by using these irreducible divisors. [ABSTRACT FROM AUTHOR]

A one to one correspondence between the elements of a finite local Frobenius non-chain ring of length 5 and nilpotency index 4, and k-tuples of DNA codewords is established. Using this map the structure of DNA codes over these rings is determined, the length of the code is relatively prime to the characteristic of the residue field of the ring.

Let $p\not =5$ be any odd prime. Using the algebraic structures of all cyclic codes of length $5p^{s}$ over the finite commutative chain ring ${\mathcal{ R}}=\mathbb F_{p^{m}}+u\mathbb F_{p^{m}}$ , in this paper, the exact values of Hamming distances of all cyclic codes of length $5p^{s}$ over $\cal R$ are established. As an application, we identify all maximum distance separable cyclic codes of length $5p^{s}$ .

Entanglement-assisted quantum error-correcting codes, which is a generalization of quantum error-correcting codes, could be derived from any classical codes by utilizing pre-shared entangled states between the sender and the receiver. In this paper, we construct some entanglement-assisted quantum maximum-distance-separable (EAQMDS) codes from constacyclic codes and cyclic codes by exploiting less pre-shared entangled states, respectively. Most of these codes are new in the sense that their parameters are not covered by the codes available in the literature. In particular, we extend the results in Tian et al. (Int. J. Theor. Phys. 60, 1843–1857, 2021) to more general case. [ABSTRACT FROM AUTHOR]

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]

Symbol-pair codes are proposed to guard against pair-errors in symbol-pair read channels. The minimum symbol-pair distance plays a vital role in determining the error-correcting capability and the constructions of symbol-pair codes with largest possible minimum symbol-pair distance is of great importance. Maximum distance separable (MDS) symbol-pair codes are optimal in the sense that such codes can acheive the Singleton bound. In this paper, for length 5p, two new classes of MDS symbol-pair codes with minimum symbol-pair distance seven or eight are constructed by utilizing repeated-root cyclic codes over F p , where p is a prime. In addition, we derive a class of MDS symbol-pair codes with minimum symbol-pair distance seven and length 4p. [ABSTRACT FROM AUTHOR]