The Constructions Of Hadamard Matrices Based On Special Binary Sequences

Hadamard matrices are square matrices consisting of ± 1,with any two rows(columns)being orthogonal.They are widely applied in many fields such as communication engineering,cryptography,and signal processing.Due to its significance,many scholars have been devoted to construct Hadamard matrices,where utilizing special binary sequences is a common method.This dissertation mainly focuses on designing special binary sequences for constructing Hadamard matrices,including a new class of sequences with four-valued autocorrelation,four sequences with two-valued autocorrelation sum and good cross-correlation sum,two classes of Goethals-Seidel(GS)sequences with even orders,and several different GS sequences by k-partitions.Each theoretical result is verified through some examples.The details are as follows.Based on the properties of finite fields,a new class of balanced binary sequences with four-valued periodic autocorrelation has been designed.By using the mapping from finite field GF(q3)to GF(q2)and the case whether the image belongs to the square element set of GF(q2)as the index set,a sequence of length q3-1 can be generated,where q is an odd prime.It is proven that the designed sequence has a period of q2+q+1,where the sequence is balanced and the periodic autocorrelation values are q2+q+1,-q,q±2.According to the theory of cyclotomic numbers,this dissertation obtains four binary sequences to construct Hadamard matrices with different standard types,where the length of four sequences is q+1/2 with a prime power q.These four sequences own the balanced properties,two-valued autocorrelation sum and good cross-correlation sum.By utilizing the sequences with these properties,the Hadamard matrices of standard type Ⅱ and Ⅳ are constructed for q ≡ 3(mod 4)and q≡1(mod 4),respectively.With the Legendre symbol and the parity properties of the associated polynomials,two methods for constructing GS sequences with length q+1 are proposed for prime power q ≡ 1(mod 4),which provides a new insight into constructing even-order GS sequences.According to sequences {ak}k=0q2-2 and {βk}k=0q2-2 in finite field GF(q),the first method is directly using {a4k}k=1q-1/2and{β4k}k=1q-1/2,while the second method is based on the parity properties of the associated polynomials of {a2k}k=0q and {β2k}k=0qThe results indicate that these two classes of GS sequences are nonequivalent.Making use of the definitions of k-blocks andk-partitions,it is proven that any four± 1 polynomials can be uniquely represented as a linear combination of the associated polynomials of an 8-block.Therefore,the construction of GS sequences can be transformed into that of 8-partitions.Finding k-partitions mainly relies on the computer search now,and however,when the order of k-partitions becomes larger,the computational expenses increase exponentially.In order to reduce the search range,some symmetry conditions of 8-partitions and 8-blocks are discussed.As a result,several nonequivalent classes of GS sequences are obtained.Additionally,one proposed 8-partition of order n can be combined with Williamson sequences with order m to generate GS sequences of order mn.Additionally,some existence conditions of k-partitions are discussed.