Combinatorial Constructions Of Geometric Orthogonal Codes With Weight Three

Nanomaterials have broad application prospects in high-tech fields such as informa-tion,biology,energy,and aerospace.DNA origami technology has important applications in the construction of nanomaterials,but deviations in DNA origami will seriously affect the properties of nanomaterials.In 2017,Doty and Winslow proposed the concept of ge-ometric orthogonal codes(GOCs),and described how to use such codes to design macro key groups in DNA origami to reduce their misalignment problems.In 2018,Chee et al.found that GOCs are closely related to optical orthogonal codes(OOCs)and optical orthogonal signature pattern codes(OOSPCs).OOCs and OOSPCs are widely used in image transmission,digital video broadcasting and other fields.Therefore,constructing a geometric orthogonal code with as many codewords as possible has important theoretical significance and practical application value.This paper mainly studies the combinatorial constructions and the existence of a class of geometric orthogonal codes with weight 3.According to the difference of ?a,we discuss(n×m,3,1)-GOC and(n×m,3,2,1)-GOC.The first part of this paper gives the number of codewords in an optimal(n×m,3,1)-GOC.On the basis of the equivalence between GOC and generalized perfect difference packing,the problems of finding the exact value of the number of codewords in a GOC is transformed into determining the number of blocks of a corresponding generalized perfect difference packing.We first construct some two-dimensional generalized perfect difference packings with difference leaves.By means of computer search,some optimal generalized(n×m,3,1)-PDPs with small parameters are given.After that,some two-dimensional optimal generalized perfect difference packings are obtained by using the filling structure,and we finally determine the exact value of the number of codewords in an optimal(n×m,3,1)-GOC.The second part of this paper studies the optimal(n×m,3,2,1)-GOCs.We first use difference methods to establish an upper bound on the number of codewords for all(n×m,3,2,1)-GOCs.Then the results on optimal(2×m,3,2,1)-GOC are established.