A note on the hull and linear complementary pair of cyclic codes

The Euclidean hull of a linear code $C$ is defined as $C\cap C^{\perp}$, where $C^\perp$ denotes the dual of $C$ under the Euclidean inner product. A linear code with zero hull dimension is called a linear complementary dual (LCD) code. A pair $(C, D)$ of linear codes of length $n$ over $\mathbb{F}_q$ is called a linear complementary pair (LCP) of codes if $C\oplus D=\mathbb{F}_q^n$. In this paper, we give a characterization of LCD and LCP of cyclic codes of length $q^m-1$, $m \geq 1$, over the finite field $\mathbb{F}_q$ in terms of their basic dual zeros and their trace representations. We also formulate the hull dimension of a cyclic code of arbitrary length over $\mathbb{F}_q$ with respect to its basic dual zero. Moreover, we provide a general formula for the dimension of the intersection of two cyclic codes of arbitrary length over $\mathbb{F}_q$ based on their basic dual zeros.