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Asymmetric quantum error-correcting codes (AQCs) may offer some advantage over their symmetric counterparts by providing better error-correction for the more frequent error types. The well-known CSS construction of <formula formulatype="inline"><tex Notation="TeX">$q$</tex> </formula>-ary AQCs is extended by removing the <formula formulatype="inline"><tex Notation="TeX">$ \BBF _{q}$</tex></formula>-linearity requirement as well as the limitation on the type of inner product used. The proposed constructions are called CSS-like constructions and utilize pairs of nested subfield linear codes under one of the Euclidean, trace Euclidean, Hermitian, and trace Hermitian inner products. After establishing some theoretical foundations, best-performing CSS-like AQCs are constructed. Combining some constructions of nested pairs of classical codes and linear programming, many optimal and good pure <formula formulatype="inline"><tex Notation="TeX">$q$</tex> </formula>-ary CSS-like codes for <formula formulatype="inline"><tex Notation="TeX">$q \in \{ 2,3,4,5,7,8,9\}$</tex> </formula> up to reasonable lengths are found. In many instances, removing the <formula formulatype="inline"> <tex Notation="TeX">$ \BBF _{q}$</tex></formula>-linearity and using alternative inner products give us pure AQCs with improved parameters than relying solely on the standard CSS construction.

CSS-Like Constructions of Asymmetric Quantum Codes