Complete Distances of All Negacyclic Codes of Length $2^{s}$ Over $\BBZ _{2^{a}}$

Various kinds of distances of all negacyclic codes of length $2^{s}$ over $\BBZ _{2^{a}}$ are completely determined. Using our structure theorems of negacyclic codes of length $2^{s}$ over $\BBZ _{2^{a}}$, we first calculate the Hamming distances of all such negacyclic codes, which particularly lead to the Hamming weight distributions and Hamming weight enumerators of several codes. These Hamming distances are then used to obtain their homogeneous, Lee, and Euclidean distances. Our techniques are extendable to the more general class of constacyclic codes, namely, the $\lambda $-constacyclic codes of length $2^{s}$ over $\BBZ _{2^{a}}$, where $\lambda $ is any unit of $\BBZ _{2^{a}}$ with the form $4k-1$. We establish the Hamming, homogeneous, Lee, and Euclidean distances of all such constacyclic codes.