Effective and reliable data retrieval is critical for the feasibility of DNA storage, and the development of random access efficiency plays a key role in its practicality and reliability. In this paper, we study the Random Access Problem, which asks to compute the expected number of samples one needs in order to recover an information strand. Unlike previous work, we took a geometric approach to the problem, aiming to understand which geometric structures lead to codes that perform well in terms of reducing the random access expectation (Balanced Quasi-Arcs). As a consequence, two main results are obtained. The first is a construction for $k=3$ that outperforms previous constructions aiming to reduce the random access expectation. The second, exploiting a result from [1], is the proof of a conjecture from [2] for rate $1/2$ codes in any dimension.

翻译：高效可靠的数据检索对于DNA存储的可行性至关重要，而随机访问效率的提升对其实际应用与可靠性起着关键作用。本文研究随机访问问题，旨在计算恢复信息链所需的预期采样次数。与先前工作不同，我们采用几何方法研究该问题，试图理解何种几何结构能够产生在降低随机访问期望值方面表现优异的编码（平衡拟弧结构）。由此获得两项主要成果：其一是针对k=3情形的构造方案，该方案在降低随机访问期望值方面超越了先前的构造方法；其二是利用文献[1]的结论，证明了文献[2]中关于任意维度下码率为1/2编码的猜想。