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The Concatenated Structure of Quasi-Abelian Codes
The decomposition of a quasi-abelian code into shorter linear codes over
larger alphabets was given in (Jitman, Ling, (2015)), extending the analogous
Chinese remainder decomposition of quasi-cyclic codes (Ling, Sol\'e, (2001)).
We give a concatenated decomposition of quasi-abelian codes and show, as in the
quasi-cyclic case, that the two decompositions are equivalent. The concatenated
decomposition allows us to give a general minimum distance bound for
quasi-abelian codes and to construct some optimal codes. Moreover, we show by
examples that the minimum distance bound is sharp in some cases. In addition,
examples of large strictly quasi-abelian codes of about a half rate are given.
The concatenated structure also enables us to conclude that strictly
quasi-abelian linear complementary dual codes over any finite field are
asymptotically good.Comment: 13 page