AI 中文总结
本文提出一种基于奇数长度循环码链的代数构造方法，无需计算机搜索即可构造具有保证自由距离和局部最小距离的卷积码，其局部码的最小距离受母码下界约束，并支持滑动窗口译码。
AI 中文摘要
本文提出了一种代数方法，基于奇数长度的循环码，无需限制即可构造具有保证的自由距离和局部最小距离的卷积码。该构造方法简单而有效，无需计算机搜索。对于任意两个正整数$r$和$t$，满足$1 \leq r < t$，可以通过使用同一长度$n$的$r$个循环码$\mathcal{C}_0, \mathcal{C}_1, \ldots, \mathcal{C}_{r-1}$的链来构造速率为$r/t$的卷积码$\mathcal{C}_{\text{convol}}$，这些循环码满足包含条件$\mathcal{C}_0 \supset \mathcal{C}_1 \supset \ldots \supset \mathcal{C}_{r-1}$。这样的卷积码$\mathcal{C}_{\text{convol}}$由限制在宽度为$n$的对角带内的半无限相同局部码链组成。$\mathcal{C}_{\text{convol}}$的每个局部码$\mathcal{C}_{\text{local}}$由码链中的$r$个循环码构成，是码链中母码$\mathcal{C}_0$的一个特殊局部化子码。$\mathcal{C}_{\text{convol}}$的每个局部码的最小距离$d_{\text{local}}$下界由码链中母码$\mathcal{C}_0$的最小距离$d_0$限定。$\mathcal{C}_{\text{convol}}$的局部结构允许基于母码$\mathcal{C}_0$的设计校验矩阵，使用滑动窗口译码方案进行译码。
英文摘要
This paper presents an algebraic method to construct convolutional codes with guaranteed \emph{free and local minimum distances} without limit based on cyclic codes of odd lengths. The constructions are simple but effective, and no computer search is needed. For any two positive integers $r$ and $t$ with $1 \leq r < t$, a rate-$r/t$ convolutional code $\mathcal{C}_{\text{convol}}$ can be constructed by using a chain of $r$ cyclic codes $\mathcal{C}_0, \mathcal{C}_1, \ldots, \mathcal{C}_{r-1}$ of the same length $n$ which satisfy the inclusion condition, $\mathcal{C}_0 \supset \mathcal{C}_1 \supset \ldots \supset \mathcal{C}_{r-1}$. Such a convolutional code $\mathcal{C}_{\text{convol}}$ is composed of a \emph{semi-infinite chain of identical local codes} confined in a diagonal band of width $n$. Each local code $\mathcal{C}_{\text{local}}$ of $\mathcal{C}_{\text{convol}}$ is formed from the $r$ cyclic codes in the code chain and is a specially localized subcode of the \emph{mother code} $\mathcal{C}_0$ in the code chain. The minimum distance $d_{\text{local}}$ of each local code of $\mathcal{C}_{\text{convol}}$ is lower bounded by the minimum distance $d_0$ of the mother code $\mathcal{C}_0$ in the code chain. The local structure of $\mathcal{C}_{\text{convol}}$ allows it to be decoded based on a designed parity-check matrix of the mother code $\mathcal{C}_0$ using a sliding window decoding scheme.
