Some necessary conditions of GS sequences based onk-partition

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 polynomials of an 8-partition. Therefore, the construction of a quad of GS sequences becomes a construction of an 8-partition. Furthermore, 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.