Quantum Convolutional Codes Derived from Generalized Reed-Solomon Codes

Abstract

Convolutional stabilizer codes promise to make quantum communication more reliable with attractive online encoding and decoding algorithms. This paper introduces a new approach to convolutional stabilizer codes based on direct limit constructions. A quantum Singleton bound for pure convolutional stabilizer codes is given. A familiy of quantum convolutional codes is derived from generalized Reed-Solomon codes. These codes are shown to be optimal with respect to the (quantum) Singleton bound.

The Road From Classical to Quantum Codes: A Hashing Bound Approaching Design Procedure

“…Both the coding efficiency and the decoding complexity of the aforementioned QCC structures are compared in Table I. Furthermore, in the spirit of finding new constructions for QCCs, Grassl et al [74], [75] constructed QCCs using the classical self-orthogonal product codes, while Aly et al explored various algebraic constructions in [76] and [77], where QCCs were derived from classical BCH codes and Reed-Solomon and Reed-Muller codes, respectively. Recently, Pelchat and Poulin made a major contribution to the decoding of QCCs by proposing degenerate Viterbi decoding [78], which runs the Maximum A Posteriori (MAP) algorithm [27] over the equivalent classes of degenerate errors, thereby improving the attainable performance.…”

New optimal quantum convolutional codes

“…In this paper, we adopt the de¯nition of memory of a quantum convolutional code according to Ref. 18, that is,…”

“…It was obtained in Ref. 18 that the free distance of a quantum convolutional code must satisfy the following version of the Singleton bound.…”

“…Grassl and R€ otteler constructed quantum convolutional codes from product codes 16 and determined an algorithm for non-catastrophic encoders. 17 Aly et al 18 established the Singleton bound for pure convolutional stabilizer codes, and constructed a class of quantum convolutional codes attaining this bound from generalized Reed-Solomon (GRS) codes. Tan and Li 19 used low-density parity-check (LDPC) codes to construct quantum convolutional codes.…”

“…However, we adopt the de¯nition of quantum convolutional memory presented in Ref. 18, since we have to compare the new code parameters with the ones shown in Ref. 18 because, one of our main results is an extension of their results.…”