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IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences

Quantum Codes Derived from Quasi-Twisted Codes of Index 2 with Hermitian Inner Product

Jingjie Lv, Ruihu Li, Qiang Fu

Published Oct 1, 2019• 5 citations
Publisher

Abstract

In this paper, we consider a wide family of λ-quasi-twisted (λ-QT) codes of index 2 and provide a bound on the minimum Hamming distance. Moreover, we give a sufficient condition for dual containing with respect to Hermitian inner product of these involved codes. As an application, some good stabilizer quantum codes over small finite fields 𝔽2 or 𝔽3 are obtained from the class of λ-QT codes.

DOI
10.1587/transfun.e102.a.1411

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In this paper, we consider a wide family of λ-quasi-twisted (λ-QT) codes of index 2 and provide a bound on the minimum Hamming distance. Moreover, we give a sufficient condition for dual containing with respect to Hermitian inner product of these involved codes. As an application, some good stabilizer quantum codes over small finite fields 𝔽2 or 𝔽3 are obtained from the class of λ-QT codes.
In this paper, a special class of two-generator quasi-twisted (QT) codes with index 2 will be presented. We explore the algebraic structure of the class of QT codes and the form of their Hermitian dual codes. A sufficient condition for self-orthogonality with Hermitian inner product is derived. Using the class of Hermitian self-orthogonal QT codes, we construct two new binary quantum codes [[70, 42, 7]]2, [[78, 30, 10]]2. According to Theorem 6 of Ref.[2], we further can get 9 new binary quantum codes. So a total of 11 new binary quantum codes are obtained and there are 10 quantum codes that can break the quantum Gilbert-Varshamov (GV) bound.
Quasi-twisted codes are used here as the classical ingredients in the so-called Construction X for quantum error-control codes. The construction utilizes nearly self-orthogonal codes to design quantum stabilizer codes. We expand the choices of the inner product to also cover the symplectic and trace-symplectic inner products, in addition to the original Hermitian one. A refined lower bound on the minimum distance of the resulting quantum codes is established and illustrated. We report numerous record breaking quantum codes from our randomized search for inclusion in the updated online database.