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

Let $F$ be a field with cardinality $p^{\mathrm{\ell }}$ and $0\mathrm{\ne }\lambda \in F$, and $0\le h<\mathrm{\ell }$. 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 $\mathrm{\ell }$ is even, the Hermitian self-dual $2$-quasi $\lambda$-constacyclic codes are asymptotically good if and only if ${\lambda }^{1+p^{\mathrm{\ell }\mathrm{/}2}}=1$. And, when $p^{\mathrm{\ell }}\equiv ̸3(\mathrm{m}\mathrm{o}\mathrm{d}4)$, the Euclidean self-dual $2$-quasi $\lambda$-constacyclic codes are asymptotically good if and only if ${\lambda }^2=1$.