  We investigate the fundamental limits of the recently proposed random access
coverage depth problem for DNA data storage. Under this paradigm, it is assumed
that the user information consists of $k$ information strands, which are
encoded into $n$ strands via some generator matrix $G$. In the sequencing
process, the strands are read uniformly at random, since each strand is
available in a large number of copies. In this context, the random access
coverage depth problem refers to the expected number of reads (i.e., sequenced
strands) until it is possible to decode a specific information strand, which is
requested by the user. The goal is to minimize the maximum expectation over all
possible requested information strands, and this value is denoted by
$T_{\max}(G)$. This paper introduces new techniques to investigate the random
access coverage depth problem, which capture its combinatorial nature. We
establish two general formulas to find $T_{max}(G)$ for arbitrary matrices. We
introduce the concept of recovery balanced codes and combine all these results
and notions to compute $T_{\max}(G)$ for MDS, simplex, and Hamming codes. We
also study the performance of modified systematic MDS matrices and our results
show that the best results for $T_{\max}(G)$ are achieved with a specific mix
of encoded strands and replication of the information strands.
