We characterize constacyclic codes in the spectral domain using the finite field Fourier transform (FFFT) and propose a reduced complexity method for the spectral-domain decoder. Further, we also consider repeated-root constacyclic codes and characterize them in terms of symmetric and asymmetric $q$-cyclotomic cosets. Using zero sets of classical self-orthogonal and dual-containing codes, we derive quantum error correcting codes (QECCs) for both constacyclic Bose-Chaudhuri-Hocquenghem (BCH) codes and repeated-root constacyclic codes. We provide some examples of QECCs derived from repeated-root constacyclic codes and show that constacyclic BCH codes are more efficient than repeated-root constacyclic codes. Finally, quantum encoders and decoders are also proposed in the transform domain for Calderbank-Shor-Steane CSS-based quantum codes. Since constacyclic codes are a generalization of cyclic codes with better minimum distance than cyclic codes with the same code parameters, the proposed results are practically useful.

翻译：我们利用有限域傅里叶变换（FFFT）在谱域中刻画常循环码，并提出一种降低复杂度的谱域译码方法。此外，我们还考虑了重根常循环码，并利用对称与非对称$q$-分圆陪集对其进行表征。通过经典自正交码与对偶包含码的零点集，我们分别推导了常循环Bose-Chaudhuri-Hocquenghem（BCH）码与重根常循环码对应的量子纠错码（QECCs）。我们给出了若干由重根常循环码构造的量子纠错码实例，并证明在相同参数下常循环BCH码比重根常循环码具有更高的效率。最后，针对基于Calderbank-Shor-Steane（CSS）构造的量子码，我们进一步提出了变换域中的量子编码器与译码器设计方案。由于常循环码是循环码的推广形式，且在相同码参数下具有比循环码更优的最小距离，本文所提出的结果具有实际应用价值。