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

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
$2\ell \mid e$
.