BCH and LCD cyclic codes of length $n=λ(q^m+1)$ with $λ\mid q-1$ are studied. A complete characterization of $q$-cyclotomic cosets modulo $n$ is given: Theorem \ref{th4} provides a necessary and sufficient condition for any $0\le γ<n$ to be a coset leader, and for odd $m$, the two largest coset leaders are explicitly determined (Theorem \ref{th9} and Theorem \ref{th14}). Based on these results, the dimensions of several families of BCH codes are determined, and the lower bound on the minimal distance of $\mathcal{C}_{(q,n,2δ+1,n-δ+1)}$ is raised to $2(δ+1)$ (Theorem \ref{th15}--\ref{th5}). Notably, several of these codes are optimal. When $m$ is odd, the necessary and sufficient condition for the BCH code $\mathcal{C}_{(q,n,δ,0)}$ to be dually-BCH is proved (Theorem \ref{th11}). Finally, an exact enumeration of all LCD cyclic codes of this length is derived (Theorem \ref{th3}). All of the above results extend previous results that were limited to $λ=1$.

翻译：本文研究了长度为 $n=λ(q^m+1)$ 且满足 $λ\mid q-1$ 的 BCH 码与 LCD 循环码。首先，给出了模 $n$ 的 $q$-分圆陪集的完整刻画：定理 \ref{th4} 为任意 $0\le γ<n$ 是陪集首元提供了充要条件，并且对于奇数 $m$，明确确定了两个最大的陪集首元（定理 \ref{th9} 与定理 \ref{th14}）。基于这些结果，确定了几类 BCH 码的维数，并将 $\mathcal{C}_{(q,n,2δ+1,n-δ+1)}$ 的最小距离下界提升至 $2(δ+1)$（定理 \ref{th15}--\ref{th5}）。值得注意的是，其中一些码是最优的。当 $m$ 为奇数时，证明了 BCH 码 $\mathcal{C}_{(q,n,δ,0)}$ 为对偶 BCH 码的充要条件（定理 \ref{th11}）。最后，推导了该长度下所有 LCD 循环码的精确计数（定理 \ref{th3}）。以上所有结果推广了先前仅限于 $λ=1$ 的研究成果。