Binary code negation and reversal are known to leave the autocorrelation sequence unchanged. However, for some code lengths, there exist code pairs for which the mechanism behind shared autocorrelation is not so simple; these are called shared-autocorrelation (or shared-ACS) binary code pairs. They are useful for, among other things, constructing binary and quad-phase Golay pairs of longer lengths. We define a symparity vector for any given binary code. For every shared-ACS binary code pairs up to length 39, the two code have the same symparity, or coupled symparity. Conditions are identified for which shared-ACS binary code pairs have coupled symparity. The coupled-symparity condition is used to develop an algorithm to find shared-ACS binary code pairs, with reduced complexity relative to more brute-force exhaustive searches. Although the algorithm assumes extra structure in the shared-ACS code pairs, it rediscovered all the sets of binary shared-ACS code pairs for lengths up to 20. It is then run for lengths 40 to 44, returning all the shared-ACS binary code pairs for each length. In particular, the algorithm results establish that there exist shared-ACS code pairs for the prime lengths 41 and 43.

中文翻译：

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利用耦合对称性的共享 ACS 二进制代码对搜索

已知二进制代码否定和反转可以保持自相关序列不变。然而，对于某些代码长度，存在共享自相关背后的机制并不那么简单的代码对；这些称为共享自相关（或共享 ACS）二进制代码对。除其他外，它们可用于构建更长长度的二元和四相 Golay 对。我们为任何给定的二进制代码定义一个对称向量。对于长度不超过 39 的每个共享 ACS 二进制代码对，这两个代码具有相同的对称性或耦合对称性。确定共享 ACS 二进制代码对具有耦合对称性的条件。耦合对称条件用于开发一种算法来查找共享 ACS 二进制代码对，相对于更强力的穷举搜索，降低了复杂性。尽管该算法在共享 ACS 代码对中假设了额外的结构，但它重新发现了长度高达 20 的所有二进制共享 ACS 代码对集。然后运行长度为 40 到 44，返回所有共享 ACS 二进制代码每个长度成对。特别是，算法结果确定存在质数长度 41 和 43 的共享 ACS 代码对。