Galois Self-Dual 2-Quasi Constacyclic Codes Over Finite Fields
Yun Fan, Yue Leng · IEEE Transactions on Information Theory · 2024
Let F be a field with cardinality$p^{\ell } $and$0 eq \lambda \in F$, 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$.