We use techniques from the fields of computer algebra and satisfiability checking to develop a new algorithm to search for complex Golay pairs. We implement this algorithm and use it to perform a complete search for complex Golay pairs of lengths up to 28. In doing so, we find that complex Golay pairs exist in the lengths 24 and 26 but do not exist in the lengths 23, 25, 27, and 28. This independently verifies work done by F. Fiedler in 2013 and confirms the 2002 conjecture of Craigen, Holzmann, and Kharaghani that complex Golay pairs of length 23 don't exist. Our algorithm is based on the recently proposed SAT+CAS paradigm of combining SAT solvers with computer algebra systems to efficiently search large spaces specified by both algebraic and logical constraints. The algorithm has two stages: first, a fine-tuned computer program uses functionality from computer algebra systems and numerical libraries to construct a list containing every sequence that could appear as the first sequence in a complex Golay pair up to equivalence. Second, a programmatic SAT solver constructs every sequence (if any) that pair off with the sequences constructed in the first stage to form a complex Golay pair. This extends work originally presented at the International Symposium on Symbolic and Algebraic Computation (ISSAC) in 2018; we discuss and implement several improvements to our algorithm that enabled us to improve the efficiency of the search and increase the maximum length we search from length 25 to 28.

我们使用计算机代数和可满足性检查领域的技术来开发一种新算法来搜索复杂的Golay对。我们实现了此算法，并使用它对长度最大为28的复数Golay对执行完整搜索。这样做时，我们发现复数Golay对存在于长度24和26中，但不存在于长度23、25，分别参见图27和28。这独立地验证了F. Fiedler在2013年所做的工作，并确认了Craigen，Holzmann和Kharaghani在2002年所做的猜想，即不存在长度为23的复数Golay对。我们的算法基于最近提出的SAT + CAS范例，该范例将SAT求解器与计算机代数系统结合起来，可以有效地搜索由代数和逻辑约束指定的大空间。该算法分为两个阶段：第一，一种经过微调的计算机程序使用计算机代数系统和数值库中的功能来构建一个列表，该列表包含每个序列，这些序列可能会在复杂的Golay对中直至等同出现为第一个序列。其次，程序化SAT求解器构造与第一阶段中构造的序列配对的每个序列（如果有），以形成复杂的Golay对。这扩展了最初在2018年国际符号和代数计算研讨会（ISSAC）上提出的工作; 我们讨论并实现了对算法的一些改进，这些改进使我们能够提高搜索效率并将搜索的最大长度从长度25增加到28。