Some necessary conditions of GS sequences based on k-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 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.
Goethals-Seidel sequencesk-partitionsymmetric or antisymmetric properties
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