MDS Codes With Galois Hulls of Arbitrary Dimensions and the Related Entanglement-Assisted Quantum Error Correction

Meng Cao Yau Mathematical Sciences Center, Tsinghua University

Information Theory mathscidoc:2205.19001

IEEE Transactions on Information Theory, 67, (12), 7964-7984, 2021.12
Let $q=p^{e}$ be a prime power and $\ell$ be an integer with $0\leq \ell\leq e-1$. The $\ell$-Galois hull of classical linear codes is a generalization of the Euclidean hull and Hermitian hull. We provide a necessary and sufficient condition under which a codeword of a GRS code or an extended GRS code belongs to its $\ell$-Galois dual code, generalizing both the Euclidean case and Hermitian case in the literature. By using four different tools: 1) the norm mapping from $\mathbb{F}_{q}^{\ast}$ to $\mathbb{F}_{p^{\ell}}^{\ast}$; 2) the direct product of two cyclic subgroups; 3) the coset decomposition of a cyclic group; 4) an additive subgroup of $\mathbb{F}_{q}$ and its cosets, we construct eleven families of $q$-ary MDS codes with $\ell$-Galois hulls of arbitrary dimensions, and give the related eleven families of $[[n,k,d;c]]_{q}$ entanglement-assisted quantum error-correcting codes (EAQECCs) with relatively large minimum distance in the sense that $2d=n-k+2+c$. We show that developing the theory of $\ell$-Galois hulls of $q$-ary MDS codes in this paper enables us to obtain new $q$-ary EAQECCs with different kinds of length sets via different $\ell$, where $2\ell\mid e$.
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@inproceedings{meng2021mds,
  title={MDS Codes With Galois Hulls of Arbitrary Dimensions and the Related Entanglement-Assisted Quantum Error Correction},
  author={Meng Cao},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220517191551418063258},
  booktitle={IEEE Transactions on Information Theory},
  volume={67},
  number={12},
  pages={7964-7984},
  year={2021},
}
Meng Cao. MDS Codes With Galois Hulls of Arbitrary Dimensions and the Related Entanglement-Assisted Quantum Error Correction. 2021. Vol. 67. In IEEE Transactions on Information Theory. pp.7964-7984. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220517191551418063258.
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