A Lattice is a partially ordered set where both least upper bound and greatest lower bound of any pair of elements are unique and exist within the set. K\"{o}tter and Kschischang proved that codes in the linear lattice can be used for error and erasure-correction in random networks. Codes in the linear lattice have previously been shown to be special cases of codes in modular lattices. Two well known classifications of semimodular lattices are geometric and distributive lattices. Most of the frequently used coding spaces are examples of either or both. We have identified the unique criterion which makes a geometric lattice distributive, thus characterizing all finite geometric distributive lattices. Our characterization helps to prove a conjecture regarding the maximum size of a distributive sublattice of a finite geometric lattice and identify the maximal case. The Whitney numbers of the class of geometric distributive lattices are also calculated. We present a few other applications of this unique characterization to derive certain results regarding linearity and complements in the linear lattice.
翻译:Lattice 是一个部分定序集, 任何元素的最小上界和最大下界都具有独特性, 并且存在于集中 。 K\ “ { o}tter ” 和 Kschisschang 证明线性线性线性线性线性线性代码可用于随机网络的错误和去除校正。 线性线性线性线性线性代码以前被显示为模块式层中代码的特殊例子。 两种已知的半模块性线性值分类是几何和分配性拉特。 大多数常用的编码空间是两者中的一种或两种。 我们已经确定了使几何性线性线性分配的独特标准, 从而将所有有限的几何分配性线性线性拉特点定性为特性。 我们的定性有助于证明关于一定几度的几何性线性线性子分解码的最大尺寸的预言, 并辨出最大值。 几何级分布性拉特特 。 也计算了几何级分布性线性线性线性线性类的Whitney 。