A pair of linear codes whose intersection is of dimension $\ell$, where $\ell$ is a non-negetive integer, is called an $\ell$-intersection pair of codes. This paper focuses on studying $\ell$-intersection pairs of $\lambda_i$-constacyclic, $i=1,2,$ and conjucyclic codes. We first characterize an $\ell$-intersection pair of $\lambda_i$-constacyclic codes. A formula for $\ell$ has been established in terms of the degrees of the generator polynomials of $\lambda_i$-constacyclic codes. This allows obtaining a condition for $\ell$-linear complementary pairs (LPC) of constacyclic codes. Later, we introduce and characterize the $\ell$-intersection pair of conjucyclic codes over $\mathbb{F}_{q^2}$. The first observation in the process is that there are no non-trivial linear conjucyclic codes over finite fields. So focus on the characterization of additive conjucyclic (ACC) codes. We show that the largest $\mathbb{F}_q$-subcode of an ACC code over $\mathbb{F}_{q^2}$ is cyclic and obtain its generating polynomial. This enables us to find the size of an ACC code. Furthermore, we discuss the trace code of an ACC code and show that it is cyclic. Finally, we determine $\ell$-intersection pairs of trace codes of ACC codes over $\mathbb{F}_4$.

翻译：摘要：两个线性码的交集维数为ℓ（ℓ为非负整数）时，称其为ℓ-交对。本文重点研究λ_i-常循环码（i=1,2）与共轭循环码的ℓ-交对。首先刻画λ_i-常循环码的ℓ-交对，基于生成多项式次数建立了ℓ的计算公式，进而得到常循环码的ℓ-线性互补对（LPC）的存在条件。随后引入并刻画有限域𝔽_{q²}上共轭循环码的ℓ-交对。研究过程中首先发现：有限域上不存在非平凡线性共轭循环码，因此重点转向加法共轭循环（ACC）码的表征。我们证明𝔽_{q²}上ACC码的最大𝔽_q-子码为循环码，并给出其生成多项式，据此可计算ACC码的规模。进一步讨论ACC码的迹码，证明其具有循环性。最终给出𝔽₄上ACC码迹码的ℓ-交对判定方法。