MDS Codes With Galois Hulls of Arbitrary Dimensions and the Related Entanglement-Assisted Quantum Error Correction
Meng Cao · IEEE Transactions on Information Theory · 2021
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$2\ell \mid e$.