  In the recent years, DNA has emerged as a potentially viable storage
technology. DNA synthesis, which refers to the task of writing the data into
DNA, is perhaps the most costly part of existing storage systems. Accordingly,
this high cost and low throughput limits the practical use in available DNA
synthesis technologies. It has been found that the homopolymer run (i.e., the
repetition of the same nucleotide) is a major factor affecting the synthesis
and sequencing errors. Quite recently, [26] studied the role of batch
optimization in reducing the cost of large scale DNA synthesis, for a given
pool $\mathcal{S}$ of random quaternary strings of fixed length. Among other
things, it was shown that the asymptotic cost savings of batch optimization are
significantly greater when the strings in $\mathcal{S}$ contain repeats of the
same character (homopolymer run of length one), as compared to the case where
strings are unconstrained.
  Following the lead of [26], in this paper, we take a step forward towards the
theoretical understanding of DNA synthesis, and study the homopolymer run of
length $k\geq1$. Specifically, we are given a set of DNA strands $\mathcal{S}$,
randomly drawn from a natural Markovian distribution modeling a general
homopolymer run length constraint, that we wish to synthesize. For this
problem, we prove that for any $k\geq 1$, the optimal reference strand,
minimizing the cost of DNA synthesis is, perhaps surprisingly, the periodic
sequence $\overline{\mathsf{ACGT}}$. It turns out that tackling the homopolymer
constraint of length $k\geq2$ is a challenging problem; our main technical
contribution is the representation of the DNA synthesis process as a certain
constrained system, for which string techniques can be applied.
