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IEEE International Symposium on Information Theory (ISIT)

Quantum Codes Derived from One-Generator Quasi-Twisted Codes

Akram Saleh, M. Reza Soleymani

Published Jul 7, 2024• 2 citations
Publisher

Abstract

In this paper, we investigate one-generator quasi-twisted (QT) codes of index 3 over finite fields. Our objective is to characterize the dual code for this class of codes in terms of Hermitian inner product. We also establish sufficient conditions for self-orthogonality of these codes with respect to this inner product. As a result of our findings, we present examples of quantum error-correcting codes (QECCs) with good and optimal parameters derived from this family of QT codes.

DOI
10.1109/isit57864.2024.10619505

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This thesis investigates the algebraic structure, construction, encoding, and decoding of quasi-cyclic (QC) and quasi-twisted (QT) codes and their applications to quantum error correction. For selected repeated-root and simple-root QC codes, generator-matrix-based construction methods are developed using the Generalized Discrete Fourier Transform (GDFT) and Discrete Fourier Transform (DFT), respectively. These methods enable efficient encoding by matrix-vector multiplication and provide computational advantages over codeword-based constructions. Spectral decoding procedures are also studied, including a GDFT-based folding framework that relates decoding of repeated-root QC codes to that of associated simple-root codes. Furthermore, a correspondence between certain repeated-root QC codes over finite fields and simple-root QT codes over finite chain rings is established. Building on these classical structures, one-generator QT codes of index 3 are investigated for quantum-code construction. Their Hermitian duals are characterized, and sufficient conditions for Hermitian self-orthogonality are derived, yielding stabilizer quantum error-correcting codes with good and, in several cases, optimal parameters. Structured syndrome-based decoding is developed by relating stabilizer measurements to classical decoding of the underlying QT codes. An extension technique preserving Hermitian self-orthogonality is introduced, producing additional stabilizer codes and corresponding decoding procedures. Finally, entanglement-assisted quantum error-correcting codes are constructed from 1-generator QT codes through Hermitian rank conditions, yielding maximal-entanglement codes with competitive parameters. Overall, the thesis establishes QC and QT codes as a unified algebraic framework for efficient classical coding, stabilizer quantum error correction, and entanglement-assisted quantum coding.
In this paper, we propose a class of 1-generator quasi-twisted codes with special structures and investigate their application to construct ternary quantum codes. We discuss the algebraic structure of these 1-generator quasi-twisted codes and their dual codes. Moreover, sufficient conditions for these quasi-twisted codes to satisfy Hermitian self-orthogonality are given. Then, some ternary quantum codes exceeding the Gilbert-Varshamov bound are derived from such Hermitian self-orthogonal 1-generator quasi-twisted codes. In particular, sixteen quantum codes are new or have better parameters than those in the literatures, eight of which are obtained by the progapation rules.