The homogeneous distance of $$(1+u^2)$$ ( 1 + u 2 ) -constacyclic codes over $$\mathbb {F}_2[u]/\langle u^3\rangle $$ F 2 [ u ] / ⟨ u 3 ⟩ and its applications

For any $$(1+u^2)$$ -constacyclic code C with arbitrary length N over $$\mathbb {F}_2[u]/\langle u^3\rangle $$ , we construct a new map $$\Phi $$ from $$(\mathbb {F}_2[u]/\langle u^3\rangle )^{N}$$ to $$(\mathbb {F}_2[v]/\langle v^2\rangle )^{2N}$$ , and derive $$\Phi (C)$$ by applying the map to each codeword of C. We find that C and its image $$\Phi (C)$$ have the same homogeneous distance, and the cyclic shift of any codeword of $$\Phi (C)$$ is still contained in $$\Phi (C)$$ over $$\mathbb {F}_2[v]/\langle v^2\rangle $$ , but $$\Phi (C)$$ is not a cyclic code in some cases. These properties help us to determine the lower bound and the upper bound for the homogeneous distance of C by discussing the same problem of $$\Phi (C)$$ . With these bounds, exact homogeneous distances of some $$(1+u^2)$$ -constacyclic codes are given. On the other hand, we consider the application of homogeneous distances in computing Hamming distances of a class of binary linear quasi-cyclic codes, and derive some optimal binary linear quasi-cyclic codes.