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MDS codes with Euclidean and Hermitian hulls of flexible dimensions and their applications to EAQECCs

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Abstract

The hull of a linear code is the intersection of itself with its dual code with respect to certain inner product. Both Euclidean and Hermitian hulls are of theorical and practical significance. In this paper, we construct several new classes of maximum distance separable (MDS) codes via (extended) generalized Reed-Solomon (GRS) codes and determine their Euclidean or Hermitian hulls. As a consequence, four new classes of MDS codes with Hermitian hulls of flexible dimensions and six new classes of MDS codes with Euclidean hulls of flexible dimensions are constructed. As applications, for the former, we further construct four new families of entanglement-assisted quantum error-correcting codes (EAQECCs) and four new families of MDS EAQECCs of length \(n>q+1\). Meanwhile, many of the distance parameters of our MDS EAQECCs are greater than \(\lceil \frac{q}{2} \rceil \) or q; for the latter, we show some examples on Euclidean self-orthogonal and one-dimensional Euclidean hull MDS codes. In addition, two new general methods for constructing extended GRS codes with \((k-1)\)-dimensional Hermitian hull and Hermitian self-orthogonal extended GRS codes are also provided.

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Acknowledgements

The authors would like to thank the editor and the anonymous referees for their helpful comments and suggestions. The authors would also like to thank the National Natural Science Foundation of China (Nos.U21A20428 and 12171134) for funding this research.

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  1. School of Mathematics, HeFei University of Technology, Hefei, 230601, China

    Yang Li, Ruhao Wan & Shixin Zhu

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  1. Yang Li
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  2. Ruhao Wan
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  3. Shixin Zhu
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Correspondence to Shixin Zhu.

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This research was supported by the National Natural Science Foundation of China (Nos.U21A20428 and 12171134).

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Li, Y., Wan, R. & Zhu, S. MDS codes with Euclidean and Hermitian hulls of flexible dimensions and their applications to EAQECCs. Quantum Inf Process 22, 153 (2023). https://doi.org/10.1007/s11128-023-03900-x

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