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.

翻译：本文针对非对称纠错问题，在有限扩域上提出了一种新的权重及相应度量。该权重能区分来自基域的元素与基域外的元素，其设计动机源于非对称量子码。我们建立了该权重与度量的理论框架，包括上下界、随机码的渐近特性，并证明存在达到Singleton型上界的最优码族。