AI 中文总结
该研究针对线性码的广义随机接入问题，建立了多符号恢复的参数上下界，推导了系统MDS、单纯形编码器的相关公式，并在三维空间中对比了平衡拟弧与基准值。
AI 中文摘要
随机接入是基于DNA的存储系统的核心需求：人们希望无需对整个编码对象进行测序即可恢复选定的信息符号。近期的组合模型将生成矩阵$G\in F_q^{k\times n}$与随机变量$\tau_i(G)$关联，该变量衡量恢复信息向量$e_i$所需的采样列数。我们研究了同时多符号恢复的基于基数的极值与有限几何方面。对于非空集合$I\subseteq[k]$，令$\tau_I(G)$表示直到所有向量$e_i$（$i\in I$）都位于观测列张成空间中所需的随机列采样数。该变量介于单符号随机接入问题和覆盖深度所基于的全恢复问题之间。对于每个$m$，我们引入大小为$m$的所有请求集合$I$上的均匀最坏情况与平均参数。利用已知的$E[\tau_I(G)]$的子集计数公式，我们为这些参数建立了一般上下界。特别地，下界通过单符号恢复变量的顺序统计量表示，且当$m=1$时可退化为已知的单符号界。对于系统MDS编码器，我们记录了已知多符号期望公式的等价形式，并推导了单调性和渐近结论。对于任意维的单纯形编码器，我们基于高斯二项式系数得到了闭式公式；全恢复端点与单纯形码的已知覆盖深度公式一致。最后，在三维空间中，我们研究了平衡拟弧并将其值与单纯形和MDS基准进行了比较。
英文摘要
Random access is a central requirement in DNA-based storage systems: one would like to recover selected information symbols without sequencing the whole encoded object. A recent combinatorial model associates to a generator matrix $G\in F_q^{k\times n}$ the random variable $τ_i(G)$, measuring the number of sampled columns needed to recover the information vector $e_i$. We study the cardinality-based extremal and finite-geometric aspects of simultaneous multi-symbol recovery. For a nonempty set $I\subseteq[k]$, let $τ_I(G)$ denote the number of random column samples needed until all vectors $e_i$, $i\in I$, lie in the span of the observed columns. This variable interpolates between the singleton random access problem and the full-recovery problem underlying coverage depth. For each $m$, we introduce uniform worst-case and average parameters over all requested sets $I$ with $|I|=m$. Using the known subset-counting formula for $E[τ_I(G)]$, we establish general upper and lower bounds for these parameters. In particular, the lower bounds are expressed through order statistics of the singleton recovery variables and specialize to the known singleton bounds when $m=1$. For systematic MDS encoders, we record an equivalent form of the known multi-symbol expectation formula and derive monotonicity and asymptotic consequences. For simplex encoders in arbitrary dimension, we obtain closed formulae in terms of Gaussian binomial coefficients; the full-recovery endpoint agrees with the known coverage-depth formula for simplex codes. Finally, in dimension three we study balanced quasi-arcs and compare their values with the simplex and MDS benchmarks.
