Abstract Pairs of complementary sequences such as Golay pairs have zero sum autocorrelation at all non-trivial phases. Several generalizations are known where conditions on either the autocorrelation function, or the entries of the sequences are altered. We aim to unify most of these ideas by introducing autocorrelation functions that apply to any sequences with entries in a set equipped with a ring-like structure which is closed under multiplication and contains multiplicative inverses. Depending on the elements of the chosen set, the resulting complementary pairs may be used to construct a variety of combinatorial structures such as Hadamard matrices, complex generalized weighing matrices, and signed group weighing matrices. We may also construct quasi-cyclic and quasi-constacyclic linear codes which over finite fields of order less than 5 are also Hermitian self-orthogonal. As the literature on binary and ternary Golay sequences is already quite deep, one intention of this paper is to survey and assimilate work on more general pairs of complementary sequences and related constructions of combinatorial objects, and to combine the ideas into a single theoretical framework.

中文翻译：

---

推广互补序列对和组合结构的构建

摘要 互补序列对（例如 Golay 对）在所有重要阶段都具有零和自相关。在自相关函数的条件或序列的条目被改变的情况下，有几种概括是已知的。我们的目标是通过引入适用于任何序列的自相关函数来统一大多数这些想法，该函数适用于具有环状结构的集合中的条目，该结构在乘法下闭合并包含乘法逆。根据所选集合的元素，产生的互补对可用于构建各种组合结构，例如 Hadamard 矩阵、复广义加权矩阵和有符号群加权矩阵。我们也可以构造准循环和准恒循环线性码，它们在小于 5 阶的有限域上也是厄米自正交的。由于关于二元和三元 Golay 序列的文献已经相当深入，本文的一个目的是调查和吸收关于更一般的互补序列对和组合对象的相关构造的工作，并将这些想法组合成一个单一的理论框架。