Skew polynomial rings over finite fields and over Galois rings have recently been used to study codes. In this paper, we extend this concept to finite chain rings. Properties of skew constacyclic codes generated by monic right divisors of $x^n-\lambda$, where $\lambda$ is a unit element, are exhibited. When $\lambda^2=1$, the generators of Euclidean and Hermitian dual codes of such codes are determined together with necessary and sufficient conditions for them to be Euclidean and Hermitian self-dual. Specializing to codes over the ring $\mathbb F$_p^m$+u\mathbb F$_p^m, the structure of all skew constacyclic codes is completely determined. This allows us to express the generators of Euclidean and Hermitian dual codes of skew cyclic and skew negacyclic codes in terms of the generators of the original codes. An illustration of all skew cyclic codes of length $2$ over $\mathbb F_3 + u\mathbb F_3$ and their Euclidean and Hermitian dual codes is also provided.

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 q-ary AQCs is extended by removing the F_q-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 q-ary CSS-like codes for q ∈ {2,3,4,5,7,8,9} up to reasonable lengths are found. In many instances, removing the F_q-linearity and using alternative inner products give us pure AQCs with improved parameters than relying solely on the standard CSS construction.

We study abelian codes in principal ideal group algebras (PIGAs). We first give an algebraic characterization of abelian codes in any group algebra and provide some general results. For abelian codes in a PIGA, which can be viewed as cyclic codes over a semisimple group algebra, it is shown that every abelian code in a PIGA admits generator and check elements. These are analogous to the generator and parity-check polynomials of cyclic codes. A characterization and an enumeration of Euclidean self-dual and Euclidean self-orthogonal abelian codes in a PIGA are given, which generalize recent analogous results for self-dual cyclic codes. In addition, the structures of reversible and complementary dual abelian codes in a PIGA are established, again extending results on reversible and complementary dual cyclic codes. Finally, asymptotic properties of abelian codes in a PIGA are studied. An upper bound for the minimum distance of abelian codes in a non-semisimple PIGA is given in terms of the minimum distance of abelian codes in semisimple group algebras. Abelian codes in a non-semisimple PIGA are then shown to be asymptotically bad, similar to the case of repeated-root cyclic codes.