Skew Constacyclic Codes over a Family of Finite Rings and Their Applications to LCD and Quantum Codes

Cyclic codes are an interesting class of linear codes due to their efficient encoding and decoding algorithms. Bose–Chaudhuri–Hocquenghem (BCH) codes which form a significant subclass of cyclic codes are important in both theory and practice since they have good error-correcting capabilities and have been widely used in communication systems, storage devices, and so on. Quantum codes with good parameters can be also constructed from BCH codes. In this paper, we construct q-ary quantum codes of length $$\frac{q^{2m}-1}{\rho }$$ using constacyclic BCH codes with order $$\rho $$ and cyclic BCH codes, respectively, where $$\rho $$ divides $$q+1$$ , q is a prime power and m is a positive integer. By comparing the obtained quantum codes, we get that constacyclic BCH codes are a better resource in constructing quantum codes than cyclic BCH codes in general. Compared with the quantum codes available in Aly et al. (IEEE Trans Inf Theory 53(3): 1183–1188, 2007) and Zhang et al. (IEEE Access 4:36122, 2018), the quantum codes in our schemes have better parameters. In particular, we extend some known results in Kai et al. (Int J Quantum Inf 16(7):1850059, 2018), La Guardia (Phys Rev A 80(4):042331, 2009), Li et al. (Quantum Inf Comput 12:0021–0035, 2013), Lin (IEEE Trans Inf Theory 50(3):5551–5554, 2004), Tang et al. (IEICE Trans Fund E102-A(1):303–306, 2019), Wang and Zhu (Quantum Inf Process 14(3):881–889, 2015), Yuan et al. (Des Codes Cryptogr 85(1):179–190, 2017) to more general case.

The theory of quantum error-correcting codes has been extended to asymmetric quantum channels-qubit-flip and phase-shift errors may have equal or different probabilities. Previous work in constructing quantum error-correcting codes has focused on code constructions for symmetric quantum channels. Recently, Galindo et al. introduced the concept of asymmetric entanglement-assisted quantum error-correcting (AEAQEC) code, and gave a Gilbert–Varshamov bound for AEAQEC codes. Then they present the explicit computation of the parameters of AEAQEC codes coming from BCH codes. In this paper, we first establish a bound for pure AEAQEC codes similar to the quantum Singleton bound, and introduce the definition of pure AEAQEC MDS codes. Then we construct three new families of AQEAEC codes by means of Vandermonde matrices, extended GRS codes and cyclic codes. The AEQAEC codes here have better parameters than the ones available in the literature.

In this paper, we study the quantum codes over $$F_{p}$$, p is an odd prime, obtained from some constacyclic codes over the finite ring $$F_{p}+uF_{p}+vF_{p}$$, where $$ u^{2}=u,v^{2}=v,uv=vu=0$$. A constacyclic code over the finite ring $$F_{p}+uF_{p}+vF_{p}$$ is decomposed into three codes over $$F_{p}$$ in order to determine the parameters of the corresponding quantum codes. Finally, we have constructed some examples of quantum error-correcting codes.

In this article, we construct some MDS quantum error-correcting codes (QECCs) from classes of constacyclic codes over R <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</sub> = F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> + u <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> + ··· + u <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</sub> F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> , u <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> = u <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> , u <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> u <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">j</sub> = u <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">j</sub> u <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> = 0, for odd prime p and i, j = 1, 2, <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">⋯</sub> , s, i ≠ j. Many QECCs with improved parameters than the existing ones in some of the earlier papers are provided. We present a set of idempotent generators of the ring Rs, and using that we define linear codes, determine all units, and study constacyclic codes over this ring. Among others, we study dual containing constacyclic codes over Rs and construct (non-binary) QECCs. An algorithm to construct QECCs from dual containing constacyclic codes over Rs is obtained that can provide many quantum codes.

Let [Formula: see text] be a prime power and let [Formula: see text] be a finite non-chain ring, where [Formula: see text], are polynomials, not all linear, which split into distinct linear factors over [Formula: see text]. We characterize constacyclic codes over the ring [Formula: see text] 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 [Formula: see text], [Formula: see text], is irrelevant while constructing quantum codes from constacyclic codes over [Formula: see text], it depends only on their degrees.

Quantum codes over finite rings have received a great deal of attention in recent years. Compared with quantum codes over finite fields, a notable advantage of quantum codes over finite rings is that they can adapt to quantum physical systems of arbitrary order. Moreover, operations are much easier to execute in finite rings than they are in fields. The modulo $m$ residue class ring $\mathbb {Z}_{m}$ is the most common finite ring. This paper investigates stabilizer codes over $\mathbb {Z}_{m}$ and presents the Gilbert–Varshamov (GV) bound. The GV bound shows that surprisingly good quantum codes exist over $\mathbb {Z}_{m}$ , which makes quantum coding feasible for arbitrary quantum physical system. We also provide an enhanced version of the GV bound for non-degenerate stabilizer codes over $\mathbb {Z}_{m}$ . The enhanced GV bound has an asymptotical form and ensures the existence of asymptotically good stabilizer codes over $\mathbb {Z}_{m}$ . Finally, these two bounds are well suited for computer searching.

&lt;abstract&gt;&lt;p&gt;Let $ s \geq 1 $ be a fixed integer. In this paper, we focus on generating cyclic codes over the ring $ \mathcal{R}(\alpha_1, \alpha_2, \ldots, \alpha_s) $, where $ \alpha_i \in \mathbb{F}_q\backslash \{0\} $, $ 1 \leq i \leq s $, by using the Gray map that is defined by the idempotents. Moreover, we describe the process to generate an idempotent by using the formula (2.1). As applications, we obtain both optimal and new quantum codes. Additionally, we solve the DNA reversibility problem by introducing $ \mathbb{F}_q $ reversibility. The aim to introduce the $ \mathbb{F}_q $ reversibility is to describe IUPAC nucleotide codes, and consequently, 5 IUPAC DNA bases are considered instead of 4 DNA bases $ (A, \; T, \; G, \; C) $.&lt;/p&gt;&lt;/abstract&gt;

Let [Formula: see text] be a commutative finite chain ring with [Formula: see text] is its maximal ideal, [Formula: see text] is its residual field and [Formula: see text] the index of nilpotency of [Formula: see text]. In this paper, we compute the dual code of some types of cyclic codes over [Formula: see text] when [Formula: see text]. Afterward, we show how extract some new quantum codes from these cyclic codes.

Let R= Fq+uFq+vFq+uvFq be a commutative ring with u2=u, v2=v, uv=vu, where q is a power of an odd prime. Ashraf and Mohammad constructed some new quantum codes from cyclic codes. Under this background, another Gray map from R to Fq4 is given. This map can be naturally extended to Rn. The problem on the ring turns to the field by this isomorphic map now. Therefore, This mapping is obviously a weight-preserving and distance-preserving map. The results show that the codes after mapping are self-orthogonal codes over Fq if they are self-orthogonal codes over R. Some computational examples show that some better non-binary quantum codes can be obtained under this Gray map. We discuss the structure of linear codes. On this basis, the structure of the generating matrix of linear codes is obtained. The structure of their dual codes is also obtained. The CSS construction guarantees the existence of quantum codes. Finally, with the help of the CSS construction, we get some good quantum codes. By comparison, our quantum codes have better parameters.

In this paper quantum codes over $\mathbb{F}_{p^{m}}$ are constructed from self-orthogonal $(1+\beta u)$-constacyclic codes of length $N=np^{s}$ over $R=\mathbb{F}_{p^{m}}+u\mathbb{F}_{p^{m}}$, $u^{2}=0$, where p is an odd prime, $(n, p)=1$, and β is a non-zero element of $\mathbb{F}_{p^{m}}$. To obtain such quantum codes, the algebraic structure of $(1+\beta u)$-constacyclic codes of length N over R is studied. The structure of their duals is also determined. From the algebraic structures of the $(1+\beta u)$-constacyclic codes and their duals over R, self-orthogonal codes over $\mathbb{F}_{p^{m}}$ are obtained as Gray images of self-orthogonal codes over R. Then applying the CalderbankShor-Steane (CSS) construction to these self-orthogonal codes over $\mathbb{F}_{p^{m}}$, the parameters of the corresponding quantum codes are determined. An example is given to illustrate the results.

Recently, La Guardia constructed some new quantum codes from cyclic codes (La Guardia, Int. J. Theor. Phys., 2017). Inspired by this work, we consider quantum codes construction from negacyclic codes, not equivalent to cyclic codes, with only one cyclotomic coset containing at least two odd consecutive integers of even length. Some new quantum codes are obtained by this class of negacyclic codes.

It is well known that n-length stabilizer quantum error correcting codes (QECCs) can be obtained via n-length classical error correction codes (CECCs) over GF(4), that are additive and self-orthogonal with respect to the trace Hermitian inner product. But, most of the CECCs have been studied with respect to the Euclidean inner product. In this paper, it is shown that n-length stabilizer QECCs can be constructed via 3n-length linear CECCs over GF(2) that are self-orthogonal with respect to the Euclidean inner product. This facilitates usage of the widely studied self-orthogonal CECCs to construct stabilizer QECCs. Moreover, classical, binary, self-orthogonal cyclic codes have been used to obtain stabilizer QECCs with guaranteed quantum error correcting capability. This is facilitated by the fact that (i) self-orthogonal, binary cyclic codes are easily identified using transform approach and (ii) for such codes lower bounds on the minimum Hamming distance are known. Several explicit codes are constructed including two pure MDS QECCs.

Let $R=\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}+\cdots+u^{k}\mathbb{F}_{2^{m}}$ , where $\mathbb{F}_{2^{m}}$ is a finite field with $2^{m}$ elements, $m$ is a positive integer, $u$ is an indeterminate with $u^{k+1}=0.$ In this paper, we propose the constructions of two new families of quantum codes obtained from dual-containing cyclic codes of odd length over $R$. A new Gray map over $R$ is defined and a sufficient and necessary condition for the existence of dual-containing cyclic codes over $R$ is given. A new family of $2^{m}$-ary quantum codes is obtained via the Gray map and the Calderbank-Shor-Steane construction from dual-containing cyclic codes over $R.$ Furthermore, a new family of binary quantum codes is obtained via the Gray map, the trace map and the Calderbank-Shor-Steane construction from dual-containing cyclic codes over $R.$

On the ℓ-DLIPs of codes over finite commutative rings

Let q = p r , where p is an odd prime and r is a positive integer. Consider a ring F q S T , where S = F q + u F q + v F q + uv F q with u 2 = 1 , v 2 = 1 , uv = vu and T = F q + u F q + v F q + w F q + uv F q + uw F q + vw F q + uvw F q with u 2 = 1 , v 2 = 1 , w 2 = 1 , uv = vu , uw = wu , vw = wv . In this paper, we examine the algebraic structure of constacyclic codes over the ring F q S T of block length ( α , β , γ ) . Further, a construction of quantum error-correcting codes (QECCs) from constacyclic codes over F q S T is also given. Moreover, we derive a number of new QECCs.