Applications of Constacyclic Codes to Some New Entanglement-Assisted Quantum MDS Codes

Generally, it is not easy to construct quantum maximal-distance-separable (MDS) codes with the minimum distance greater than
$\frac {q}{2}+1$
. The minimum distance of quantum MDS codes can achieve
$\frac {q}{2}+1$
or exceed
$\frac {q}{2}+1$
by adopting pre-shared entanglement. In this work, some new families of entanglement-assisted quantum MDS codes that satisfy the quantum Singleton bound are constructed and the number of maximally entangled states required is determined to make the minimum distance of some constructed codes achieve
$\frac {q}{2}+1$
or exceed
$\frac {q}{2}+1$
by utilizing the decomposition of the defining set and
$q^{2}$
-cyclotomic cosets of constacyclic codes with length
$\frac {q^{2}+1}{\gamma }$
, where
$\gamma =t^{2}+1$
,
$t$
is a power of 2 and
$q=t^{e}>4$
with
$e\equiv 1~\textrm {mod}~4$
or
$e\equiv 3~\textrm {mod}~4$
. Moreover, the parameters of these codes constructed in this paper are more general relative to the ones in the literature and the minimum distance of some codes constructed in this paper is larger than
$\frac {q}{2}+1$
.