几何正交码的界与组合构造

Obtaining new nanomaterials to meet people's needs has become a hot topic in scientific research. DNA origami plays an important role in the construction of nanomaterials. The deviation of DNA origami will affect the characteristics of nanomaterials. Therefore, Doty introduced geometric orthogonal code (GOC) to solve the macro dislocation problem in DNA origami. The number of geometric orthogonal codewords corresponds to the number of macrobonds that can be processed in the corresponding origami region, the weight of the code word of geometric orthogonal code corresponds to the macrostructure strength, and different macrostructure strength can improve the efficiency when constructing nanomaterials compared with the single macrobond strength. Therefore, for a given weight set W, it is significant to construct geometric orthogonal codes with as many code words as possible.This project aims to study geometric orthogonal codes via combinatorial design and finite field, we mainly focus on the following problems : (1) Systematic constructions optimal (n,4,1)-GOCs, and partially determines the exact value of the number of code words in (n,4,1)-GOCs; （2)the bound and construction of (n,w,2)-GOCs; (3) give the upper bound of (n,W,λ,Q)-GOCs, and construct optimal (n,W,1,Q)-GOCs.

获取新纳米材料以满足人们的需求已成为当今科学研究的一个热点问题。DNA折纸技术在构造纳米材料中发挥了巨大的作用。DNA折纸中如果出现偏差将会影响纳米材料的特性，Doty等人引入几何正交码(简记为GOC)以解决DNA折纸中出现的宏建错位问题。几何正交码码字个数对应于折纸区域能够处理的宏建数目，码字重量对应于宏建强度，有差异的宏建强度比单一宏建强度能提高构造纳米材料的效率。因此，对于给定重量集W，构造出码字个数尽可能多的几何正交码很有意义。本项目利用组合设计、有限域等研究几何正交码，主要涉及以下问题:(1)系统构造最优(n,4,1)-GOCs，部分确定(n,4,1)-GOC码字个数精确值；(2)(n,w,2)-GOC的界与构造；(3)给出(n,W,λ,Q)-GOC的码字个数上界，构造最优(n,W,1,Q)-GOC码类。

本项目主要研究以下内容：1.最优几何正交码(GOC)及与其相关的最优光正交签名码(OOSPC)的界与构造；2.区组大小为4的可分(decomposable)超单平衡不完全区组设计(BIBD)和可分组设计(GDD)的存在性；3.无理完美幻方(Irrational MPMS)的存在性。在几何正交码与光正交签名码方面，对一般参数给出了变重量几何正交码的上界，并构造了一些最优码类；对任意正整数 m,n,构造了最优(6m,6n,4,1)-OOSPCs和最优(2m,18n,4,1)-OOSPCs。在可分超单设计方面，完全解决了λ=4,6,8时可分超单(v,4,λ)-BIBDs的存在性；对λ=2,4，除了2个可能例外，解决了λ-可分超单(4,λ)-GDD(g^u)的存在性。彻底解决了无理完美幻方的存在性。本项目已发表论文4篇，均为SCI源刊，招收在读硕士一人。