Narrow Results

New families of subsystem codes are constructed. Furthermore, these new families of subsystem codes are optimal. Finally, new families of asymmetric subsystem codes which derived from cyclic codes are presented.

By using q²-ary cyclotomic cosets, we construct new asymmetric quantum codes with parameters [[q²+1, q²+1-2(k+i-2), (2k+3)/(2i+3)]]_q², where k and i are positive integers, and . The constructed asymmetric quantum codes are not covered by the codes available in the literature. Moreover, the constructed asymmetric quantum codes are optimal.

In this paper, we construct two classes of asymmetric quantum codes by using constacyclic codes. The first class is the asymmetric quantum codes with parameters [[q² + 1, q² + 1 - 2(t + k + 1), (2k + 2)/(2t + 2)]]_q² where q is an odd prime power, t, k are integers with , which is a generalization of [J. Chen, J. Li and J. Lin, Int. J. Theor. Phys. 53 (2014) 72, Theorem 2] in the sense that we do not assume that q ≡1 (mod 4). The second one is the asymmetric quantum codes with parameters , where q ≥ 5 is an odd prime power, t, k are integers with 0 ≤ t ≤ k ≤ q - 1. The constructed asymmetric quantum codes are optimal and their parameters are not covered by the codes available in the literature.

Most of quantum codes have been constructed by using classical linear codes over finite field. However, little is known about the construction of quantum codes via group character codes. In this work, we present the constructions of nonbinary quantum and asymmetric quantum codes based on group character codes, and give explicit parameters for infinite families of nonbinary quantum codes and asymmetric quantum codes. Furthermore, we present families of optimal quantum codes which derived from group character codes.

By a careful analysis on cyclotomic cosets, the maximal designed distance δ_new of narrow-sense imprimitive Euclidean dual containing q-ary BCH code of length is determined, where q is a prime power and l is odd. Our maximal designed distance δ_new of dual containing narrow-sense BCH codes of length n improves upon the lower bound δ_max for maximal designed distances of dual containing narrow-sense BCH codes given by Aly et al. [IEEE Trans. Inf. Theory53 (2007) 1183]. A series of non-narrow-sense dual containing BCH codes of length n, including the ones whose designed distances can achieve or exceed δ_new, are given, and their dimensions are computed. Then new quantum BCH codes are constructed from these non-narrow-sense imprimitive BCH codes via Steane construction, and these new quantum codes are better than previous results in the literature.

The entanglement-assisted (EA) formalism generalizes the standard stabilizer formalism. All quaternary linear codes can be transformed into entanglement-assisted quantum error correcting codes (EAQECCs) under this formalism. In this work, we discuss construction of EAQECCs from Hermitian non-dual containing primitive Bose–Chaudhuri–Hocquenghem (BCH) codes over the Galois field GF(4). By a careful analysis of the cyclotomic cosets contained in the defining set of a given BCH code, we can determine the optimal number of ebits that needed for constructing EAQECC from this BCH code, rather than calculate the optimal number of ebits from its parity check matrix, and derive a formula for the dimension of this BCH code. These results make it possible to specify parameters of the obtained EAQECCs in terms of the design parameters of BCH codes.

Maximal-entanglement entanglement-assisted quantum error-correcting codes (EAQE-CCs) can achieve the EA-hashing bound asymptotically and a higher rate and/or better noise suppression capability may be achieved by exploiting maximal entanglement. In this paper, we discussed the construction of quaternary zero radical (ZR) codes of dimension five with length $n\ge 5$. Using the obtained quaternary ZR codes, we construct many maximal-entanglement EAQECCs with very good parameters. Almost all of these EAQECCs are better than those obtained in the literature, and some of these EAQECCs are optimal codes.

The Bose–Chaudhuri–Hocquenghem (BCH) codes have been studied for more than 57 years and have found wide application in classical communication system and quantum information theory. In this paper, we study the construction of quantum codes from a family of $q^2$-ary BCH codes with length $n=q^{2m}+1$ (also called antiprimitive BCH codes in the literature), where $q\ge 4$ is a power of 2 and $m\ge 2$. By a detailed analysis of some useful properties about $q^2$-ary cyclotomic cosets modulo $n$, Hermitian dual-containing conditions for a family of non-narrow-sense antiprimitive BCH codes are presented, which are similar to those of $q^2$-ary primitive BCH codes. Consequently, via Hermitian Construction, a family of new quantum codes can be derived from these dual-containing BCH codes. Some of these new antiprimitive quantum BCH codes are comparable with those derived from primitive BCH codes.

By studying the properties of $q^2$-cyclotomic cosets, the maximum designed distances of Hermitian dual-containing constacyclic Bose–Chaudhuri–Hocquenghem (BCH) codes with length $n=2(q^m+1)$ are determined, where $q$ is an odd prime power and $m\ge 3$ is an integer. Further, their dimensions are calculated precisely for the given designed distance. Consequently, via Hermitian Construction, many new quantum codes could be obtained from these codes, which are not covered in the literature.

Hermitian dual-containing codes are an important class of linear codes which have important applications in the construction of quantum codes. In this paper, two classes of Hermitian dual-containing almost MDS codes over finite fields are studied. By employing the Hermitian construction, a class of quantum codes with minimum distance $3$ and a class of quantum codes with minimum distance $4$ are constructed.

In this paper, we study nonbinary quantum codes from cyclic codes over the ring $F_p[u]/〈u^3-u〉$ for odd prime $p$. We give the constructional properties of linear and cyclic codes over the ring $F_p[u]/〈u^3-u〉$. As an applications of these classes of codes, we obtain quantum codes over $F_p$ using self-orthogonal property. Moreover, we find some new nonbinary quantum codes with better parameters than the known quantum codes over $F_p$.

Let $(R,uR,k,e)$ be a commutative finite chain ring with $uR$ is its maximal ideal, $k$ is its residual field and $e$ the index of nilpotency of $u$. In this paper, we compute the dual code of some types of cyclic codes over $R$ when $e=3$. Afterward, we show how extract some new quantum codes from these cyclic codes.