  DNA storage has emerged as an important area of research. The reliability of
DNA storage system depends on designing the DNA strings (called DNA codes) that
are sufficiently dissimilar. In this work, we introduce DNA codes that satisfy
a special constraint. Each codeword of the DNA code has a specific property
that any two consecutive sub-strings of the DNA codeword will not be the same
(a generalization of homo-polymers constraint). This is in addition to the
usual constraints such as Hamming, reverse, reverse-complement and
$GC$-content. We believe that the new constraint will help further in reducing
the errors during reading and writing data into the synthetic DNA strings. We
also present a construction (based on a variant of stochastic local search
algorithm) to calculate the size of the DNA codes with all the above
constraints, which improves the lower bounds from the existing literature, for
some specific cases. Moreover, a recursive isometric map between binary vectors
and DNA strings is proposed. Using the map and the well known binary codes we
obtain few classes of DNA codes with all the constraints including the property
that the constructed DNA codewords are free from the hairpin-like secondary
structures.
