On the complete weight enumerators of some reducible cyclic codes. (English) Zbl 1321.94129
Summary: Let \(\mathbb{F}_r\) be a finite field with \(r = q^m\) elements and \(\theta\) a primitive element of \(\mathbb{F}_r\). Suppose that \(h_1(x)\) and \(h_2(x)\) are the minimal polynomials over \(\mathbb{F}_q\) of \(g_1^{- 1}\) and \(g_2^{- 1}\), respectively, where \(g_1, g_2 \in \mathbb{F}_r^\ast\). Let \(\mathcal{C}\) be a reducible cyclic code over \(\mathbb{F}_q\) with check polynomial \(h_1(x) h_2(x)\). In this paper, we investigate the complete weight enumerators of the cyclic codes \(\mathcal{C}\) in the following two cases: (1) \(g_1 = \theta^{\frac{q - 1}{h}}, g_2 = \beta g_1\), where \(h > 1\) is a divisor of \(q - 1\), \(e > 1\) is a divisor of \(h\), and \(\beta = \theta^{\frac{r - 1}{e}}\); (2) \(g_1 = \theta^2, g_2 = \theta^{p^k + 1}\), where \(q = p\) is an odd prime and \(k\) is a positive integer. Moreover, we explicitly present the complete weight enumerators of some cyclic codes.
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