In the past few decades, design theory has grown to encompass a wide variety of research directions. It comes as no surprise that applications in coding theory and communications continue to arise, and also that designs have found applications in new areas. Computer science has provided a new source of applications of designs, and simultaneously a field of new and challenging problems in design theory. In this paper, we revisit a construction for orthogonal designs using the multiplication tables of Cayley-Dixon algebras of dimension 2 n . The desired orthogonal designs can be described by a system of equations with the aid of a Gröbner basis computation. For orders greater than 16 the combinatorial explosion of the problem gives rise to equations that are unfeasible to be handled by traditional search algorithms. However, the structural properties of the designs make this problem possible to be tackled in terms of rewriting techniques, by equational unification. We establish connections between central concepts of design theory and equational unification where equivalence operations of designs point to the computation of a minimal complete set of unifiers. These connections make viable the computation of some types of orthogonal designs that have not been found before with the aforementioned algebraic modeling.

中文翻译：

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通过符号计算和重写技术构建 2 的幂的正交设计

在过去的几十年里，设计理论已经发展到涵盖了广泛的研究方向。毫不奇怪，编码理论和通信中的应用不断涌现，而且设计也已在新领域找到了应用。计算机科学提供了设计应用的新来源，同时也提供了设计理论中新的和具有挑战性的问题领域。在本文中，我们使用维数为 2 n 的 Cayley-Dixon 代数的乘法表重新审视正交设计的构造。所需的正交设计可以在 Gröbner 基计算的帮助下通过方程组进行描述。对于大于 16 的阶数，问题的组合爆炸会产生传统搜索算法无法处理的方程。然而，设计的结构特性使这个问题可以通过等式统一的重写技术来解决。我们在设计理论的中心概念和等式统一之间建立联系，其中设计的等价运算指向计算最小完整统一集。这些连接使得某些类型的正交设计的计算变得可行，这些类型之前在上述代数建模中没有发现。