Combinatorial Constructions Of Geometric Orthogonal Codes

DNA origami is a new DNA self-assembly method proposed in recent years,which is widely used in DNA computation and nanotechnology.In 2017,Doty and Winslow introduced geometric orthogonal codes(GOCs)and geometric 180-rotating orthogonal codes(RGOCs)to design macrobonds in 3D DNA origami,so as to reduce misalignments and mismatches.Doty and Chee et al.also found that geometric orthogonal codes are closely related to optical orthogonal codes(OOCs)and optical orthogonal signature codes(OOSPCs).They presented the first class of GOCs with more codewords than the optimal OOCs under the same parameters.So far,the research on geometric orthogonal codes and geometric 180-rotating orthogonal codes is still in the initial stage.The known results on GOCs and RGOCs are only as follows.The preliminary upper and lower bounds on the number of codewords are given.A class of geometric orthogonal codes and a class of geometric 180-rotating orthogonal codes are obtained using polynomials over finite fields.And the existence results of the optimal geometric orthogonal codes with the codeword weight 3 and the correlation coefficient 1 are obtained using the combinatorial configuration,called geometric difference packings(GDPs).In this paper,we mainly use some auxiliary designs in combinatorics to construct the infinite classes of variable-weight geometric orthogonal codes.The existence of two types of variable-weight geometric orthogonal codes is completely determined with a few possible exceptional values.In addition,we obtain the exact size of a class of optimal geometric orthogonal codes using the method of set theory.For geometric 180-rotating orthogonal codes,we provide a general upper bound on the codewords capacity and determine the exact value of the codewords capacity for a class of optimal codes.This paper is organized as follows:In Chapter 1,we give the combinatorial descriptions of(n × m,k,λ)-GOCs and(n × m,k,λ)-RGOCs,and provide general upper bounds on the number of codewords of these two kinds of codes.In particular,we determine the codewords capacities of optimal(n × m,k,-1)-GOCs and optimal(n ×m,k-1)-RGOCs for any positive integers n,m and k.In Chapter 2,we establish the recursive constructions of GOCs and RGOCs,and the construction methods from OOCs and OOSPCs to GOCs.As results,we get some new infinite classes of(n×m,k,λ)-GOCs and(n ×m,k,λ)-RGOCs for λ<01.In Chapter 3,we discuss the existences of(n ×m,K,1)-geometric difference families(GDFs)for K={3,4} and K={3,4,5}.According to the equivalent relationship between variable-weight GOCs with the correlation coefficient of 1 and GDFs,the existence of perfect(n × m,{3,4},1)-GOCs is completely solved by constructing GDFs.And the existence of perfect(n×m,{3,4,5},1)-GOCs is almost completely determined with a few possible exceptional values.In Chapter 4,we consider the codewords capacity of optimal(n×m,k,λ_a,k-1)GOCs for λ_a ≤k-2.We introduce the definition based on set theory and equivalent combinatorial descriptions of this kind of geometric orthogonal codes.The correlation coefficient is connected with the multiplicity of differences in the difference list.By analyzing the distribution of elements in the difference list,we determine the exact value of the optimal code capacity for λ_a=k-2 or λ_a=k-3,and give the calculation methods of the exact value of the optimal code capacity for λ_a≤k-4.In Chapter 5,we consider the codewords capacity of optimal(n ×m,k,k-2,k-1)RGOCs.By characterizing the codewords structure of this type of RGOC and providing its equivalent combination description,we determine the exact value of the optimal code capacity for any positive integers n,m and k.In Chapter 6,we summarize the main conclusions of this paper and list the further research questions.