Quantum Constacyclic BCH Codes over Qudits: A Spectral-Domain Approach

TL;DR

This paper introduces a spectral-domain approach to analyze and decode quantum constacyclic BCH codes over qudits, offering reduced complexity methods and new quantum error-correcting code constructions.

Contribution

It characterizes constacyclic codes in the spectral domain, proposes a reduced complexity spectral decoder, and derives new quantum error-correcting codes from these codes.

Findings

Constacyclic BCH codes are more efficient than repeated-root constacyclic codes.

Spectral-domain characterization simplifies decoding processes.

New QECCs are constructed with improved parameters.

Abstract

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.