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Construction of quantum codes from multivariate polynomial rings

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In this paper, we use multivariate polynomial rings to construct quantum error-correcting codes (QECCs) via Hermitian construction. We establish a relation between linear codes and ideals of multivariate polynomial rings. We give a necessary and suffcient condition for a multivariate polynomial to generate a Hermitian dual-containing code. By comparing with the literatures in recent years, we construct 31 new QECCs over \(\mathbb {F}_q\), where \(q=3,4,5,7\). Some of them reach quantum singleton bound and some of them exceed quantum GV bound.

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Data Availability Statement

The data that support the findings of this study are available upon request from the corresponding author. Restrictions apply to the availability of these data, which were used under license for this study and thus are not publicly available. Data are however available from the authors upon reasonable request and with permission of the data provider.


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We would like to express our sincere appreciation to all those who have contributed to this paper. This work was supported by the National Natural Science Foundation of China under Grant Nos U21A20428 and 12171134.

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Correspondence to Shixin Zhu.

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We declare that this paper represents original work and has not been submitted, in whole or in part, for publication elsewhere. All sources used in the preparation of this manuscript have been properly cited. The work reported in this paper was conducted within the framework of appropriate ethical conduct, and any potential conflicts of interest have been disclosed. We take full responsibility for the content and conclusions presented in this paper.

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Yu, C., Zhu, S. & Tian, F. Construction of quantum codes from multivariate polynomial rings. Des. Codes Cryptogr. (2024).

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