论文标题
通过差异三角集建造LDPC卷积代码
Construction of LDPC convolutional codes via difference triangle sets
论文作者
论文摘要
在本文中，在任意有限领域的$（N，K，δ）$ LDPC卷积代码的结构概括了Robinson和Bernstein的工作以及后来的TONG工作。形成$（k，w）$ - （弱的）差异三角集的整数集用作支撑$（n，k，Δ）$卷积代码的滑动平价检查矩阵的某些列，其中$ n \ in \ mathbb {n} $，$ n> k $。卷积代码的参数与基础差异三角集的参数有关。特别是，建立了代码的自由距离与$ w $之间的关系，以及代码程度和差异三角集范围之间的关系。此外，我们表明，弱差三角集合的某些条件可确保与卷积代码的滑动奇偶校验检查矩阵相关的坦纳图从$ 2 \ ell $ -cycles中免费，无法满足任何有限字段的整个等级条件。最后，我们放松这些条件，并根据$ \ ell $的奇偶校验提供野外大小的下限，这足以避免$ 2 \ ell $ -Cycles。这对于改善代码的性能并避免存在低重量的代码字和吸收套件很重要。
In this paper, a construction of $(n,k,δ)$ LDPC convolutional codes over arbitrary finite fields, which generalizes the work of Robinson and Bernstein and the later work of Tong is provided. The sets of integers forming a $(k,w)$-(weak) difference triangle set are used as supports of some columns of the sliding parity-check matrix of an $(n,k,δ)$ convolutional code, where $n\in\mathbb{N}$, $n>k$. The parameters of the convolutional code are related to the parameters of the underlying difference triangle set. In particular, a relation between the free distance of the code and $w$ is established as well as a relation between the degree of the code and the scope of the difference triangle set. Moreover, we show that some conditions on the weak difference triangle set ensure that the Tanner graph associated to the sliding parity-check matrix of the convolutional code is free from $2\ell$-cycles not satisfying the full rank condition over any finite field. Finally, we relax these conditions and provide a lower bound on the field size, depending on the parity of $\ell$, that is sufficient to still avoid $2\ell$-cycles. This is important for improving the performance of a code and avoiding the presence of low-weight codewords and absorbing sets.
