基于k-划分构造GS序列的一些必要条件
Some necessary conditions of GS sequences based on k-partition
沈淑慧1
作者信息
- 1. 电子科技大学数学科学学院,四川 成都 611731
- 折叠
摘要
Goethals-Seidel(GS)序列是四个由±1元素组成的序列组,它在Hadamard矩阵的构造中有着重要的作用,即将一组GS序列直接嵌入到GS阵列中.一种常见的构造GS序列的方法是利用k-划分,因此有必要讨论某些k-划分存在的必要条件.首先,任意四个±1序列关联多项式可以等价地写成一组8-划分关联多项式的线性组合,这种线性组合形式是唯一的,基于此GS序列的构造可以等价地转换为一组8-划分的构造.进一步利用 k-划分和GS序列的定义得到一个该k-划分存在的必要条件:若使用具有对称或者反对称性质的k-划分来构造GS序列,则序列的长度必须为偶数.
Abstract
The Goethals-Seidel(GS)sequences consisting of±1 play a crucial role in constructing Hadamard matrices by plugging a quad of GS sequences into GS arrays.To construct GS sequences,a common method is utilizing k-partitions,and thus it is meaningful to discuss some necessary conditions for the existence of k-partitions.It is first proven that any four associated polynomials of±1 sequences can be equivalently expressed as a unique linear combination of the associated polyno-mials of an 8-partition.Therefore,the construction of a quad of GS sequences becomes a construction of an 8-partition.Fur-thermore,utilizing the definition of k-partitions and GS sequences,it is deduced that the length of k-partitions with symmetric or antisymmetric properties has to be even.
关键词
Goethals-Seidel序列/k-划分/对称和反对称Key words
Goethals-Seidel sequences/k-partition/symmetric or antisymmetric properties引用本文复制引用
基金项目
国家自然科学基金(61771004)
国家自然科学基金(6237109)
出版年
2024
NETL
NSTL
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