Galois Self-Dual 2-Quasi Constacyclic Codes Over Finite Fields

Abstract

Let F be a field with cardinality $p^{\ell } $ and $0\neq \lambda \in F$ , and $0\le h\lt \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 $p^{h}$ -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 $p^{\ell } \,{\cancel {\equiv }}\,3~({\mathrm { mod}}~4)$ , the Euclidean self-dual 2-quasi $\lambda $ -constacyclic codes are asymptotically good if and only if $\lambda ^{2}=1$ .