弱弧及其在基于DNA的存储访问问题中的应用
Weak arcs and applications to the DNA-based storage access problem
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中文总结 AI 辅助
本文研究PG$(n-1,q)$中的弱弧及其平衡变体，给出弱弧大小上界、刻画平面最大平衡拟弧并在PG$(3,q)$中构造大平衡拟弧，将其用于构建基于DNA存储随机访问问题的点集，构造显式且恢复期望匹配最佳渐近界。
中文摘要 AI 辅助
弱弧是PG$(n-1,q)$中的点集，其与每个不经过标准基向量所给点的一般超平面的交点数最多为$n-1$个。本文研究弱弧以及包含在基本单形边上的平衡变体，给出了弱弧大小的上界，刻画了平面中最大的平衡拟弧，并在PG$(3, q)$中构造了大的平衡拟弧。随后利用这些构型为基于DNA的存储中的随机访问问题构建点集，这些构造是显式的，适用于小域，且达到的恢复期望匹配已知的最佳渐近界。
英文摘要
Weak arcs are point sets in PG$(n-1,q)$ meeting every general hyperplane (those are the hyperplanes not going through one of the points given by the standard basis vectors) in at most $n-1$ points. In this paper, we study weak arcs together with balanced variants which are contained on the sides of the fundamental simplex. We give an upper bound on the size of weak arcs, characterise the largest balanced quasi-arcs in the plane and construct large balanced quasi-arcs in PG$(3, q)$. We then use these configurations to build point sets for the random-access problem in DNA-based storage. The constructions are explicit, work over small fields, and attain recovery expectations matching the best known asymptotic bounds.
