Semi-cyclic Holey Group Divisible Designs With Block Size Four And Their Application

An optical orthogonal code is a family of sequence with good autocorrelation and cross-correlation and are widely used in code-division multiple access systems(CDMA).At present,most of the research on optical orthogonal code at home and abroad focus on the optimal capacity and construction of optical orthogonal code,while the semi-cyclic holey group divisible design(SCHGDD)has an important application in the construction of optical orthogonal code.The necessary and sufficient conditions for the existence of 3-SCHGDD of type(n,mt)have been fully solved by scholars,while the results for 4-SCHGDD of type(n,mt)are rare.In this paper,we study the existence of 4-SCHGDD of type(n,mt)and apply the results in the study of optical orthogonal codes.The definition of semi-cyclic holey group divisible design comes from holey group divisible design(HGDD),using the method of pure difference and mix difference to define the k-HGDD of type(n,mt),when the pure difference and mix difference to meet the corresponding conditions,so the HGDD is called SCHGDD.First in chapter 1 briefly introduce the research status of SCHGDD and basic concepts,and briefly describe the main results obtained in this paper.Chapter 2 examines the presence of 4-SCHGDD of type(n,mt).Using the cyclic holey difference matrix(CHDM)and SCHGDD,4-SCHGDD and base groups are obtained;when m∈{2,4,6,8} several 4-SCHGDD of type(4,mt)are determined.Using recursive construction and the necessary and sufficient conditions for the existence of 4-SCGDD of type mn,we obtain sufficient conditions for the existence of 4-SCHGDD of type(n,mt).Chapter 3 investigates the presence conditions of the perfect AM-OPPSP/WP 3-D(u×v×w,4,1)-OOC.Using the existential relationship between kSCHGDD of type(u,wv)and w-cyclic HGDD of type(u,wv),combined with the equivalent relationship between HGDD and AM-OPPSP/WP 3-D OOC,it is to apply the existence of 4-SCHGDD of type(u,wv),we obtain the sufficient condition for the existence of perfect AM-OPPSP/WP 3-D(u×v×w,4,1)-OOC.Chapter 4 proves that the original root of pα when p is a prime is also the original root of p.Finally in chapter 5,the full text is summarized and the future research work is prospected.