The primitive idempotents of irreducible constacyclic codes and LCD cyclic codes

Abstract

In this paper, we investigate all irreducible factors of $x^{l_{1}^{m_{1}}l_{2}^{m_{2}}} - a$ over $\mathbb {F}_{q}$ and obtain all primitive idempotents in $\mathbb {F}_{q}[x]/\langle x^{l_{1}^{m_{1}}l_{2}^{m_{2}}} - a \rangle$, where $a \in \mathbb {F}_{q}^{*}$, l₁, l₂ are two distinct odd prime divisors of q^t − 1 with $\gcd (l_{1}l_{2},q(q-1))= 1$ for prime t. Furthermore, the weight distributions of all irreducible constacyclic codes of length $l_{1}^{m_{1}}l_{2}^{m_{2}}$ are presented for t = 2. As an application, we determine all linear complementary dual cyclic codes of length $l_{1}^{m_{1}}l_{2}^{m_{2}}$ over $\mathbb {F}_{q}$.