Trace description and Hamming weights of irreducible constacyclic codes

Irreducible constacyclic codes constitute an important family of error-correcting codesand have applications in space communications.In this paper, we provide a trace description of irreducible constacyclic codes of length $n$ over the finite field $\mathbb{F}_{q}$ of order $q,$ where $n$ is a positive integer and $q$ is a prime power coprime to $n.$ As an application, we determine Hamming weight distributions of some irreducible constacyclic codes of length $n$ over $\mathbb{F}_{q}.$ We also derive a weight-divisibility theorem for irreducible constacyclic codes, and obtain both lower and upper bounds on the non-zero Hamming weights in irreducible constacyclic codes. Besides illustrating our results with examples, we list some optimal irreducible constacyclic codes that attain the distance bounds given in Grassl's Table [8].