AI 中文总结
本文提出递归构造的Fractus稀疏矩阵，用于生成具有近线性编码复杂度和低解码复杂度的LDPC码，保证围长六和最小距离，并建立序理论与编码理论的联系。
AI 中文摘要
我们引入了一族新的递归构造的稀疏矩阵，称为 Fractus 矩阵，并研究它们在构造低密度奇偶校验（LDPC）码中的应用。这些矩阵通过自相似递归过程生成，在连续迭代中保持关键结构性质的同时，产生正则稀疏校验矩阵。这种递归结构使得编码算法具有近乎线性的计算复杂度（与块长度成线性关系）。解码采用标准的迭代消息传递算法，从而保留了 LDPC 码低复杂度解码的特性。所提出的构造产生的 Tanner 图围长为六，并保证最小汉明距离至少为 $\ell+1$。我们建立了 Fractus 矩阵的若干代数性质，包括稀疏性、正则性、递归分解以及翻转转置操作下的对称性。此外，我们表明 Fractus 矩阵族具有自然的格结构，并且相关的 LDPC 码继承了相应的格论性质。这些结果建立了序理论与编码理论之间的联系。总体而言，所提出的框架将递归矩阵构造、高效编码、图论分析和格论整合为一种统一的代数方法，用于可扩展 LDPC 码的设计与分析。
英文摘要
We introduce a new family of recursively constructed sparse matrices, termed Fractus matrices, and investigate their use in constructing low-density parity-check (LDPC) codes. Generated through a self-similar recursive process, these matrices yield regular sparse parity-check matrices while preserving key structural properties across successive iterations. This recursive structure enables an efficient encoding algorithm with computational complexity that is nearly linear in the block length. Decoding is performed using standard iterative message-passing algorithms, thereby retaining the low-complexity decoding characteristic of LDPC codes. The proposed construction produces Tanner graphs with girth six and guarantees a minimum Hamming distance of at least $\ell+1$. We establish several algebraic properties of Fractus matrices, including sparsity, regularity, recursive decomposition, and symmetry under the flip-transpose operation. In addition, we show that the family of Fractus matrices admits a natural lattice structure and that the associated LDPC codes inherit corresponding lattice-theoretic properties. These results establish a connection between order theory and coding theory. Overall, the proposed framework integrates recursive matrix constructions, efficient encoding, graph-theoretic analysis, and lattice theory into a unified algebraic approach to the design and analysis of scalable LDPC codes.
Comments43 pages
