Quantum error-correcting codes play the role of suppressing noise and decoherence in quantum systems by introducing redundancy. Some strategies can be used to improve the parameters of these codes. For example, entanglement can provide a way for quantum error-correcting codes to achieve higher rates than the one obtained by means of the traditional stabilizer formalism. Such codes are called entanglement-assisted quantum error-correcting (EAQEC) codes. In this paper, we utilize algebraic geometry codes to construct several families of EAQEC codes derived from the Euclidean and the Hermitian construction. Three families constructed here consist of codes whose quantum Singleton defect is equal to zero, one, or two. We also construct families of EAQEC codes with an encoding rate exceeding the quantum Gilbert-Varshamov bound. Additionally, asymptotically good towers of linear complementary dual codes are used to obtain asymptotically good families of EAQEC codes consuming maximal entanglement. Furthermore, a simple comparison with the quantum Gilbert-Varshamov bound demonstrates that, by utilizing the proposed construction, it is possible to generate an asymptotically family of EAQEC codes that exceeds this bound.

Quantum error correcting codes play the role of suppressing noise and decoherence in quantum systems by introducing redundancy. Some strategies can be used to improve the parameters of these codes. For example, entanglement can provide a way for quantum error correcting codes to achieve higher rates than the one obtained via traditional stabilizer formalism. Such codes are called entanglement-assisted quantum (QUENTA) codes. In this paper, we use algebraic geometry codes to construct two families of QUENTA codes, where one of them has maximal entanglement and is maximal distance separable. In the end, we show that for any asymptotically good tower of algebraic function fields there is an asymptotically good family of maximal entanglement QUENTA codes with nonzero rate, relative minimal distance, and relative amount of entanglement.

In this paper, we construct new families of classical convolutional codes (CCC's) and new families of quantum convolutional codes (QCC's). The CCC's are derived from (block) algebraic geometry (AG) codes. Furthermore, new families of CCC's are constructed by applying the techniques of puncturing, extending, expanding, and by the direct product code construction applied to AG codes. In addition, utilizing the new CCC's constructed here, we obtain new families of QCC's. The parameters of these new codes are good. More precisely, in the classical case, a family of almost near maximum distance separable (MDS) codes is presented; in the quantum case, we construct a family of MDS (optimal) quantum convolutional codes.

Several new families of multi-memory classical convolutional Bose-Chaudhuri-Hocquenghem 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.

Four quantum code constructions generating several new families of good nonbinary quantum nonprimitive nonnarrow-sense, Bose-Chaudhuri-Hocquenghem codes, are presented in this paper. The first two are based on Calderbank-Shor-Steane (CSS) construction derived from two nonprimitive Bose-Chaudhuri-Hocquenghem codes. The third one is based on Steane's enlargement of nonbinary CSS codes applied to suitable subfamilies of nonprimitive nonnarrow-sense Bose-Chaudhuri-Hocquenghem codes. The fourth construction is derived from suitable subfamilies of Hermitian dual-containing nonprimitive nonnarrow-sense Bose-Chaudhuri-Hocquenghem codes. These constructions generate new families of quantum codes whose parameters are better than the ones available in the literature.

New asymmetric quantum generalized Reed-Solomon (RS) codes with parameters [[N = mn, K = m(2k - n + c), dz≥ d/dx≥ [d - c)]]q, where q is a prime power, 1m, k = n - d + 1, and n, d >; c + 1, c ≥ 1, m ≥ 1 are integers, are constructed in this paper. These new codes can be utilized in quantum channels having great asymmetry.

New quantum maximum-distance-separable (MDS) codes with parameters [[q2+ 1,q2-2d+ 3,d]]q, whereq=2t,t≥ 1 and 3 ≤d≤q+1 is an odd integer, are constructed in this paper.