  Linear programming approaches have been applied to derive upper bounds on the
size of classical codes and quantum codes. In this paper, we derive similar
results for general quantum codes with entanglement assistance, including
nonadditive codes, by considering a type of split weight enumerators. After
deriving the MacWilliams identities for these split weight enumerators, we are
able to prove algebraic linear programming bounds, such as the Singleton bound,
the Hamming bound, and the first linear programming bound. In particular, we
show that the first linear programming bound improves the Hamming bound when
the relative distance is sufficiently large.
  On the other hand, we obtain additional constraints on the size of Pauli
subgroups for quantum codes, which allow us to improve the linear programming
bounds on the minimum distance of small quantum codes. In particular, we show
that there is no [[27,15,5]] or [[28,14,6]] quantum stabilizer code. We also
discuss the existence of some entanglement-assisted quantum stabilizer codes
with maximal entanglement. As a result, the upper and lower bounds on the
minimum distance of maximal-entanglement quantum stabilizer codes with length
up to 20 are significantly improved.
