Constructions of Asymmetric Quantum Alternant Codes

Abstract

Asymmetric quantum error-correcting codes (AQCs) have been proposed to deal with the significant asymmetry in many quantum channels, which may have more flexbility than general quantum error-correcting codes (QECs). In this paper, we construct AQCs based on Alternant codes. Firstly, we propose a new subclass of Alternant codes and combine them with BCH codes to construct AQCs. Then we construct AQCs based on series of nested pairs of subclasses of Alternant codes such as nested Goppa codes. As an illustrative example, we get three [[55, 6, 19/4]], [[55, 10, 19/3]], [[55, 15, 19/2]] AQCs from the well known [55, 16, 19] binary Goppa code. At last, we get asymptotically good binary expansions of quantum GRS codes, which are quantum generalizations of Retter’s classical results.