介绍了基于π-旋转矩阵的低密度奇偶校验码(LDPC码)的构造方法,对π-旋转矩阵进行了研究和改造,对其约束条件进行加强,在此基础上定义了Q-矩阵,并提出Q-矩阵LDPC码的构造方法.Q-矩阵是约束满足问题的解,具有快速搜索算法,并能用循环移位的方法获得Q-矩阵集.利用Q-矩阵能快速灵活地构造不合4线循环的大型稀疏奇偶校验矩阵,从而生成LDPC码.提出的编码器设计基本思想是按照有利于LDPC码的构成及其电路设计的方式将奇偶校验矩阵H分解成两个子矩阵,通过对H的分解与重构运算直接构成码字.由于不需生成矩阵G,使LDPC码编码器的实现代价大幅度的降低.
Mixed states are of interest in quantum mechanics for modelling partial information. More recently categorical approaches to linguistics have also exploited the idea of mixed states to describe ambiguity and hyponym / hypernym relationships. In both these application areas the category Rel of sets and binary relations is often used as an alternative model. Selinger's CPM construction provides the setting for mixed states in Hilbert space based categorical quantum mechanics. By analogy, applying the CPM construction to Rel is seen as introducing mixing into a relational setting. We investigate the category CPM(Rel) of completely positive maps in Rel. We show that the states of an object in CPM(Rel) are in bijective correspondence with certain families of graphs. Via map-state duality this then allows us provide a graph theoretic characterization of the morphisms in CPM(Rel). By identifying an appropriate composition operation on graphs, we then show that CPM(Rel) is isomorphic to a category of sets and graphs between them. This isomorphism then leads to a graph based description of the complete join semilattice enriched dagger compact structure of CPM(Rel). These results allow us to reason about CPM(Rel) entirely in terms of graphs. We exploit these techniques in several examples. We give a closed form expression for the number of states of a finite set in CPM(Rel). The pure states are characterized in graph theoretic terms. We also demonstrate the possibly surprising phenomenon of a pure state that can be given as a mixture of two mixed states.
由于BP算法中的非线性运算较复杂，实现中通常采用Min—Sum近似简化译码算法。针对译码过程中需要存储大量信息的问题，本文提出了一种基于Min—Sum近似算法的QC—LDPC译码器。通过重新安排Min—Sum近似算法中的运算，并将校验节点信息以一种压缩冗余的形式表示，大大减少了译码器所需的存储空间。针对QC—LDPC码校验矩阵准循环的特性，译码过程中以块为单位对信息进行更新，且可以实现多种消息传递调度策略。为进一步减少存储空间，对变量节点信息采用了非线性量化，根据密度演进理论对量化规则进行了优化。
We study the eigenvector problem in homogeneous superintegrable
N-state chiral Potts model (CPM) by the symmetry principal. Using duality symmetry and (spin-)inversion in CPM, together with Onsager-algebra symmetry and
sl2-loop-algebra symmetry of the superintegrable
τ(2)-model, we construct the complete
k′-dependent CPM-eigenvectors in the local spin basis for an arbitrary Onsager sector. In this paper, we present the complete classification of quantum numbers of superintegrable
τ(2)-model. Accordingly, there are four types of sectors. The relationships among Onsager sectors under duality and inversion, together with their Bethe roots and CPM-eigenvectors, are explicitly found. Using algebraic-Bethe-ansatz techniques and duality of CPM, we construct the Bethe states and the Fabricius-McCoy currents of the superintegrable
τ(2)-model through its equivalent spin-
2N−1-XXZ chain. The
τ(2)-eigenvectors in a sector are derived from the Bethe state and the
sl2-product structure determined by the Fabricius-McCoy current of the sector. From those
τ(2)-eigenvectors, the
k′-dependence of CPM state vectors in local-spin-basis form is obtained by the Onsager-algebra symmetry of the superintegrable chiral Potts quantum chain.
We study binary quasi-cyclic LDPC codes whose parity-check matrices are full arrays of single circulant permutation matrices (CPMs), referred to here as CPM-LDPC codes. Their minimum distance is at most
(J+1)!, where
J is the column weight. For every fixed pair of column and row weights $2\le J
围长（girth）较大的Qc-LDPC码，由于译码性能优良而且便于硬件实现，因此目前已经成为国际信道编码领域的一个研究热点。最近，J—WZhang在第四届多媒体信息网络与安全国际会议上提出了一种构造girth-8（3，L）QC-LDPC码的新方案，但是没有对围长特性和分块矩阵尺寸取值进行理论分析和论证。利用等价变换和最大公约数体系，本文得到了该方案的围长特性和分块矩阵尺寸取值的精确理论结果。此外，还利用最大公约数体系对该方案进行了扩展，得到一种新的围长为8的QC—LDPC码。新码的参数选取范围包含很多原方案不适用的参数选择范围。仿真结果说明，新码在和积译码算法下具有较优良的译码性能。