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Four quantum code constructions generating several new families of good nonbinary quantum nonprimitive non-narrow-sense Bose-Chaudhuri-Hocquenghem (BCH) codes are presented in this paper. The first two ones are based on Calderbank-Shor-Steane (CSS) construction derived from two nonprimitive BCH codes, not necessarily self-orthogonal. The third one is based on nonbinary Steane's enlargement of CSS codes applied to suitable sub-families of nonprimitive non-narrow-sense BCH codes. The fourth construction is derived from suitable sub-families of Hermitian self-orthogonal nonprimitive non-narrow-sense BCH codes. These constructions generate new families of quantum BCH codes whose parameters are better than the ones available in the literature

New families of classical and quantum optimal negacyclic convolutional codes are constructed in this paper. This optimality is in the sense that they attain the classical (quantum) generalized Singleton bound. The constructions presented in this paper are performed algebraically and not by computational search.

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Abstract-Several new families of multi-memory classical convolutional Bose-Chaudhuri-Hocquenghem (BCH) codes as well as families of unit-memory quantum convolutional codes are constructed in this paper. Our unit-memory classical and quantum convolutional codes are optimal in the sense that they attain the classical (quantum) generalized Singleton bound. The constructions presented in this paper are performed algebraically and not by computational search.Index Terms -convolutional codes, quantum convolutional codes, MDS codes, cyclic codes I. INTRODUCTION Several works available in the literature deal with constructions of quantum error-correcting codes (QECC) [4-9, 13, 17, 21, 22, 24, 33, 41, 42]. In contrast with this subject of research one has the theory of quantum convolutional codes [1-3, 12, 14-16, 34, 35, 43-45]. Ollivier and Tillich [34,35] were the first to develop the stabilizer structure for these codes. Almeida and Palazzo Jr. construct an [(4, 1, 3)] (memory m = 3) quantum convolutional code [1]. constructed quantum convolutional codes as well as they provide algorithms to obtain non-catastrophic encoders. Forney, in a joint work with Guha and Grassl, constructed rate (n − 2)/n quantum convolutional codes. Wilde and Brun [44,45] constructed entanglement-assisted quantum convolutional coding and Tan and Li [43] constructed quantum convolutional codes derived from LDPC codes.Constructions of (classical) convolutional codes and their corresponding properties as well as constructions of optimal convolutional codes (in the sense that they attain the generalized Singleton bound [38]) have been also presented in the literature [11,18,25,28,29,36,[38][39][40]. In particular, in the paper by Rosenthal and York [39], the authors obtained some of the matrices of the state-space realization of the convolutional codes in the same way as the parity check matrix of a BCH block code, generating convolutional codes with different structures of (classical block) BCH codes. As it is well known, the generalized (classical) Singleton bound [38] (see also [40]) appears recently in the literature. In the paper by Piret [37] and even in the handbook [36], the concept of MDS convolutional codes was addressed, but in a different context that the previously mentioned. In this paper we use the notion of MDS convolutional codes according to Smarandache and Rosenthal [40].Keeping these facts in mind, in this paper we propose constructions of new families of quantum and classical convolutional codes by applying the famous method proposed by

A family of asymmetric quantum codes derived from the product code of two (classical) Reed–Solomon (RS) codes is constructed in this paper. This family is constructed by applying the Calderbank–Shor–Steane (CSS) construction. The proposed codes can be utilized in quantum channels having great asymmetry, that is, quantum channels in which the probability of occurrence of phase-shift errors is large when compared to the probability of occurrence of qudit-flip errors.

In this paper, we construct several new families of quantum codes with good parameters. These new quantum codes are derived from (classical) t-point (t ≥ 1) algebraic geometry (AG) codes by applying the Calderbank-Shor-Steane (CSS) construction. More precisely, we construct two classical AG codes C1 and C2 such that C1 ⊂ C2, applying after the well-known CSS construction to C1 and C2. Many of these new codes have large minimum distances when compared with their code lengths as well as they also have small Singleton defects. As an example, we construct a family [[46, 2(t2 − t1), d]] 25 of quantum codes, where t1, t2 are positive integers such that 1 < t1 < t2 < 23 and d ≥ min{46 − 2t2, 2t1 − 2}, of length n = 46, with minimum distance in the range 2 ≤ d ≤ 20, having Singleton defect at most four. Additionally, by applying the CSS construction to sequences of t-point (classical) AG codes constructed in this paper, we generate sequences of asymptotically good quantum codes. *