Galois Self-dual 2-quasi Constacyclic Codes over Finite Fields

Let $F$ be a field with cardinality $p^\ell$ and $0\neq \lambda\in F$, and $0\le h<\ell$. Extending Euclidean and Hermitian inner products, Fan and Zhang introduced Galois $p^h$-inner product (DCC, vol.84, pp.473-492). In this paper, we characterize the structure of $2$-quasi $\lambda$-constacyclic codes over $F$; and exhibit necessary and sufficient conditions for $2$-quasi $\lambda$-constacyclic codes being Galois self-dual. With the help of a technique developed in this paper, we prove that, when $\ell$ is even, the Hermitian self-dual $2$-quasi $\lambda$-constacyclic codes are asymptotically good if and only if $\lambda^{1+p^{\ell/2}}=1$. And, when $p^\ell\,{\not\equiv}\,3~({\rm mod}~4)$, the Euclidean self-dual $2$-quasi $\lambda$-constacyclic codes are asymptotically good if and only if $\lambda^{2}=1$.