A finite sequence of numbers is perfect if it has zero periodic autocorrelation after a nontrivial cyclic shift. In this work, we study quaternionic perfect sequences having a one-to-one correspondence with the binary sequences arising in Williamson's construction of quaternion-type Hadamard matrices. Using this correspondence, we devise an enumeration algorithm that is significantly faster than previously used algorithms and does not require the sequences to be symmetric. We implement our algorithm and use it to enumerate all circulant and possibly non-symmetric Williamson-type matrices of orders up to 21; previously, the largest order exhaustively enumerated was 13. We prove that when the blocks of a quaternion-type Hadamard matrix are circulant, the blocks are necessarily pairwise amicable. This dramatically improves the filtering power of our algorithm: in order 20, the number of block pairs needing consideration is reduced by a factor of over 25,000. We use our results to construct quaternionic Hadamard matrices of interest in quantum communication and prove they are not equivalent to those constructed by other means. We also study the properties of quaternionic Hadamard matrices analytically, and demonstrate the feasibility of characterizing quaternionic Hadamard matrices with a fixed pattern of entries. These results indicate a richer set of properties and suggest an abundance of quaternionic Hadamard matrices for sufficiently large orders.

翻译：若有限数字序列在非平凡循环移位后具有零周期自相关性，则称其为完美序列。本文研究四元数完美序列，这类序列与Williamson构造四元数型哈达玛矩阵时产生的二进制序列存在一一对应关系。利用该对应关系，我们设计了一种枚举算法，其速度显著快于现有算法，且不要求序列具有对称性。我们实现了该算法，并借此枚举了所有阶数不超过21的循环型及可能非对称的Williamson型矩阵；此前已穷举验证的最大阶数为13。我们证明了当四元数型哈达玛矩阵的分块为循环矩阵时，这些分块必然两两亲和。这一结论极大提升了算法的筛选能力：在20阶情形中，需要考察的分块对数量减少了超过25,000倍。我们利用所得结果构造了量子通信领域感兴趣的四元数哈达玛矩阵，并证明它们与其他方法构造的矩阵不等价。我们还通过解析方法研究了四元数哈达玛矩阵的性质，论证了以固定元素模式表征四元数哈达玛矩阵的可行性。这些结果表明四元数哈达玛矩阵具有更丰富的性质集，并预示着在足够大的阶数下存在大量此类矩阵。