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MDS Codes With Galois Hulls of Arbitrary Dimensions and the Related Entanglement-Assisted Quantum Error Correction.
- Source :
- IEEE Transactions on Information Theory; Dec2021, Vol. 67 Issue 12, p7964-7984, 21p
- Publication Year :
- 2021
-
Abstract
- 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 on $\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 $2\ell \mid e$. [ABSTRACT FROM AUTHOR]
- Subjects :
- ERROR-correcting codes
QUANTUM entanglement
LINEAR codes
INTEGERS
CYCLIC groups
Subjects
Details
- Language :
- English
- ISSN :
- 00189448
- Volume :
- 67
- Issue :
- 12
- Database :
- Complementary Index
- Journal :
- IEEE Transactions on Information Theory
- Publication Type :
- Academic Journal
- Accession number :
- 153731625
- Full Text :
- https://doi.org/10.1109/TIT.2021.3117562