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

TL;DR

The paper tackles QECC design for qudit systems by extending constacyclic BCH codes to the spectral domain via the finite-field Fourier transform, enabling efficient syndrome-based and zero-set–driven decoding in transform space.It introduces a reduced-complexity spectral decoding algorithm and provides a detailed encoding/decoding architecture based on Q(FFFT), with explicit syndrome extraction circuits and Clifford-encoding mappings.The authors establish both weakly self-dual and dual-containing quantum constacyclic BCH codes, and extend the framework to repeated-root constacyclic codes, showing that constacyclic BCH codes generally achieve better parameters than their repeated-root counterparts.Overall, the work delivers practical, transform-domain QECC constructions with encoding/decoding circuits, offering potential efficiency gains for qudit-based quantum information processing.

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.

Figures (2)

• Figure 1: Encoding circuit for quantum constacyclic BCH codes. The encoding operator $\mathcal{E} = Q^{-1}$ is performed on the first $n$ qudits of the initial state $\ket{\psi_0}$ to obtain the codeword $\ket{\psi}$. $\ket{\epsilon}$ is obtained from $\ket{0}$ by performing $\mathrm{DFT}_{p^{k'}}^{-1}$. Then the initial state $\ket{\psi_0}$ is obtained from message state $\ket{\phi}$, $\ket{0}$s, and $\ket{\epsilon}$s.
• Figure 1: Syndrome computation circuit for quantum constacyclic BCH codes. The $n_m$, $n_{\mathrm{X}}$, and $n_{\mathrm{Z}}$ represent the number of qudits corresponding to subblocks $D_M$, $D_{\mathrm{X}}$, and $D_{\mathrm{Z}}$, respectively. The syndrome computation involves performing $Q= \mathrm{Q}(\mathrm{FFFT})$, $\mathrm{ADD}_{p^{k'}}$, $\mathrm{DFT}_{p^{k'}}$, $\mathrm{DFT}_{p^{k'}}^{-1}$, and $Q^{-1}$ operations on the codeword qudits along with $2(\delta-1)$ syndrome qudits that are initially in the state $\ket{0}$ each. At the end of the syndrome computation, the syndrome qudits are in the state $\ket{s} = \ket{s_{\mathrm{X}}}\otimes \ket{s_{\mathrm{Z}}}$, while the state of the codeword qudits is $E\ket{\psi}$. $\ket{v_1}$, $\ket{v_2}$, $\ket{v_3}$, and $\ket{v_4}$ represent the intermediate states of the codeword qudits along with the syndrome qudits.

Theorems & Definitions (58)

• Definition 1
• Theorem 1: wan2003lectures, Theorem 10.7
• Definition 2
• Lemma 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof
• Theorem 4