图上的编码：可观测性、可控性和局部可约性
Codes on Graphs: Observability, Controllability and Local Reducibility
AI总结：
本文研究一般图上线性或群码实现的性质，通过引入修剪性和恰当性的对偶概念，证明最小实现与局部可约性的关系，并给出可观测性和可控性的计数测试及其与局部可约性的联系。
摘要：
本文研究一般图上线性或群码实现的导致局部可约性的性质。修剪性和恰当性是约束码的对偶性质。如果线性或群实现的约束码不是既修剪又恰当的，则它是局部可约的。在有限无环图上，线性或群实现是最小的当且仅当每个局部约束码都是修剪且恰当的。如果码字与配置之间存在一一对应，则实现称为可观测的；如果具有独立约束，则称为可控的。线性或群实现是可观测的当且仅当其偶像是可控的。给出了可控性的简单计数测试。不可观测或不可控的实现是局部可约的。奇偶校验实现是可控制的当且仅当它们具有独立的奇偶校验。在不可控的尾咬网格实现中，行为划分为不连通的子行为，但这一性质对于非网格实现不成立。在一般图上，不可观测配置的支撑是一个广义环。
英文摘要：
This paper investigates properties of realizations of linear or group codes on general graphs that lead to local reducibility. Trimness and properness are dual properties of constraint codes. A linear or group realization with a constraint code that is not both trim and proper is locally reducible. A linear or group realization on a finite cycle-free graph is minimal if and only if every local constraint code is trim and proper. A realization is called observable if there is a one-to-one correspondence between codewords and configurations, and controllable if it has independent constraints. A linear or group realization is observable if and only if its dual is controllable. A simple counting test for controllability is given. An unobservable or uncontrollable realization is locally reducible. Parity-check realizations are controllable if and only if they have independent parity checks. In an uncontrollable tail-biting trellis realization, the behavior partitions into disconnected subbehaviors, but this property does not hold for non-trellis realizations. On a general graph, the support of an unobservable configuration is a generalized cycle.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
