  We introduce a new weight and corresponding metric over finite extension
fields for asymmetric error correction. The weight distinguishes between
elements from the base field and the ones outside of it, which is motivated by
asymmetric quantum codes. We set up the theoretic framework for this weight and
metric, including upper and lower bounds, asymptotic behavior of random codes,
and we show the existence of an optimal family of codes achieving the
Singleton-type upper bound.
