Generalized concatenated codes and quasi-abelian codes over chain rings: Distance bounds and asymptotic goodness

In this paper, we investigate generalized concatenated codes over finite commutative chain rings and derive two lower bounds on their distances with respect to an arbitrary weight function. One bound generalizes Jensen’s lower bound, while the other extends a recent bound obtained by Özbudak and Özkaya (2024), with the latter proving to be as effective as the former and, under certain conditions, even superior. We also provide a trace description for all quasi-abelian codes over finite commutative chain rings, offering a construction method for these codes. Using this description, we show that quasi-abelian codes can be viewed as generalized concatenated codes. Additionally, we derive the generalized concatenated structure of the Euclidean dual code of each quasi-abelian code. We also derive two lower bounds on the homogeneous distances of quasi-abelian codes over finite commutative chain rings, using their generalized concatenated structure. As applications, we demonstrate that quasi-abelian codes and their special class consisting of Euclidean LCD codes are asymptotically good with respect to the homogeneous metric, establishing the existence of two asymptotically good classes of linear codes over finite commutative chain rings.