大部分的研究學者在描述超立方體系統時都會使用節點位址表示法（address notation），然而，這種表示法有一些缺點，如較沒有邏輯上的意義或不方便作運算等。我們於本篇論文中提出一個全新的超立方體表示法—超立方體代數（hypercube algebra）。由於它與布林代數的相似性非常高，在我們定義了它的運算子、運算元及公設與定理之後，我們可以更輕易的使用它來幫助我們表示與處理超立方體的相關問題。
此外，在較大型的超立方體系統中，元件（節點與鍊結）錯誤的機率也相對的提高，如何保留可用的完整超立方體來有效的達到效能緩慢降低（graceful de-gradation）的目標。在本論文中我們也針對主要次立方體的判定（prime subcubes determination）問題提出三個解決的演算法，分別為超立方體函數（hypercube function）、Q-map（Q對應圖）以及同步訊息傳送（SMP）演算法。而這三種演算法也都是源自於超立方體代數的概念。

Most of previous researches use node address as the notation for hypercube systems. However, this notation has many shortages such as: too trivial, complicated symbolic representation, no logical structural meaning. This thesis presents a new algebra system, called hypercube algebra, which can be used as a logical notation for hypercubes as well as for incomplete hypercubes. Due to the similarity between hypercube algebra and Boolean algebra, cubeterms (i.e. similar to minterm in Boolean algebra), sum and product operations (i.e., similar to sum and product in Boolean algebra), and certain axioms are defined for hypercube algebra. Using hypercube algebra, a simple hypercube expression can carry many topological properties in a hypercube system such as prime cubes, node availability and so on.
On the other hand, the probability of fault occurrence can be high for a large hypercube system. It is often desire to reconfigure the faulty hypercube that operate in a gracefully degraded manner as to retain many nonfaulty nodes and links as possible, by supporting the execution of parallel algorithms in smaller nonfaulty subcubes. Therefor the subcube determination problem is essential that the time for executing a parallel algorithm trends to depend on the dimension of the assigned subcube. Here, we also present three different methods which are hypercube function, Q-map, and synchronized message passing (SMP) method for determining subcube according different uses and situations. The basic ideas of these three methods are from the hypercube algebra.

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