Complete weight enumerators of some cyclic codes

Abstract

Let ${\mathbb {F}}_r$ be a finite field with $r=q^m$ elements, $\alpha$ a primitive element of ${\mathbb {F}}_r$, $\hbox {Tr}_{r/q}$ the trace function from ${\mathbb {F}}_r$ onto ${\mathbb {F}}_q$, $r-1=nN$ for two integers $n, N \ge 1$, and $\theta =\alpha ^N$. In this paper, we use Gauss sums to investigate the complete weight enumerators of irreducible cyclic codes

$$\begin{aligned} {{\mathcal {C}}}=\big \{{\mathbf{c }}(a)=(\hbox {Tr}_{r/q}(a), \hbox {Tr}_{r/q}(a\theta ), \ldots , \hbox {Tr}_{r/q}\big (a\theta ^{n-1}\big ): a \in {\mathbb {F}}_r\big \} \end{aligned}$$

and explicitly present the complete weight enumerators of some irreducible cyclic codes when $\gcd (n, q-1)=q-1 \text{ or } \frac{q-1}{2}$. Moreover, we determine the complete weight enumerators of a class of cyclic codes with the check polynomials $h_1(x)h_2(x)$ by using Gauss sums, where $h_i(x)$ are the minimal polynomials of $\alpha _i^{-1}$ over ${\mathbb {F}}_q$ and ${\mathbb {F}}_{q^{m_i}}^*=\langle \alpha _i \rangle$ for $i=1,2$. We shall obtain their explicit complete weight enumerators if $\gcd (m_1,m_2)=1$ and $q=3$ or $4$.