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arXiv:1612.07157 · IEEE Transactions on Communications

Classical and Quantum Convolutional Codes Derived From Algebraic Geometry Codes

Francisco Revson F. Pereira, Giuliano G. La Guardia, Francisco de Assis

Published Oct 12, 2018• 9 citations
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Abstract

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.

DOI
10.1109/tcomm.2018.2875754

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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.
Quantum maximum-distance-separable (MDS) codes form an important family of quantum codes. Quantum MDS codes are optimal, which can attain the quantum Singleton bound. In this paper, using some classes of constacyclic codes and Hermitian constructions, we propose the constructions of optimal quantum convolutional codes and optimal asymmetric quantum codes in the sense of the quantum Singleton bound. Compared with the codes obtained in the literature, the parameters of the constructed quantum codes are new as well as the distance dz of optimal asymmetric quantum codes [[n,k,dz/dx]]q2 constructed in this paper is larger.
In this paper we construct several new families of quantum codes with good and asymptotically good parameters. These new quantum codes are derived from (classical) algebraic geometry (AG) codes by applying the Calderbank-Shor-Steane (CSS) construction. Many of these codes have large minimum distances when compared with its code length and they have relatively small Singleton defect. For example, we construct a family [[46; 2(t_2 - t1); d]]_25 of quantum codes, where t_1, t_2 are positive integers such that 1 < t_1 < t_2 < 23 and d >= min {46 - 2t_2, 2t_1-2}, of length n = 46, with minimum distance in the range 2 =< d =< 20, having Singleton defect four. Additionally, by utilizing t-point AG codes, with t >= 2, we show how to obtain sequences of asymptotically good quantum codes.
In this paper, we construct the first families of asymmetric quantum convolutional codes (AQCC)'s. These new AQCC's are constructed by means of the CSS-type construction applied to suitable families of classical convolutional codes, which are also constructed here. The new codes have noncatastrophic generator matrices and they present great asymmetry. Since our constructions are performed algebraically, it is possible to derive several families of such codes and not only codes with specific parameters. Additionally, several different types of such codes are obtained.
Quantum convolutional codes can be used to protect a sequence of qubits of arbitrary length against decoherence. We introduce two new families of quantum convolutional codes. Our construction is based on an algebraic method which allows to construct classical convolutional codes from block codes, in particular BCH codes. These codes have the property that they contain their Euclidean, respectively Hermitian, dual codes. Hence, they can be used to define quantum convolutional codes by the stabilizer code construction. We compute BCH-like bounds on the free distances which can be controlled as in the case of block codes, and establish that the codes have non-catastrophic encoders.