Factorizations of Binomial Polynomials and Enumerations of LCD and Self-Dual Constacyclic Codes

Constacyclic codes are well-known generalizations of cyclic and negacyclic codes. Due to their rich algebraic structure, constacyclic codes are used to construct quantum codes and symbol-pair codes. Let
${\mathbb {F}}_{q}$
be a finite field with order
$q$
, where
$q$
is a positive power of a prime
$p$
. Suppose that
$n$
is a positive integer and the product of distinct prime factors of
$n$
divides
$q-1$
, i.e.,
$rad(n)\mid (q-1)$
. In this paper, we explicitly factorize the polynomial
$x^{n}-\lambda $
for each
$\lambda \in {\mathbb {F}}_{q}^{*}$
. As applications, first, we obtain all repeated-root
$\lambda $
-constacyclic codes and their dual codes of length
$np^{s}$
over
${\mathbb {F}}_{q}$
; second, we determine all simple-root LCD cyclic codes and LCD negacyclic codes of length
$n$
over
${\mathbb {F}}_{q}$
; third, we list all self-dual repeated-root negacyclic codes of length
$np^{s}$
over
${\mathbb {F}}_{q}$
. In contrast to known results, the lengths of constacyclic codes in this paper have more flexible parameters.