DNA codes for additive stem similarity

We study two new concepts of combinatorial coding theory: additive stem similarity and additive stem distance between $q$-ary sequences. For $q=4$, the additive stem similarity is applied to describe a mathematical model of thermodynamic similarity, which reflects the “hybridization potential” of two DNA sequences. Codes based on the additive stem distance are called DNA codes. We develop methods to prove upper and lower bounds on the rate of DNA codes analogous to the well-known Plotkin upper bound and random coding lower bound (the Gilbert–Varshamov bound). These methods take into account both the “Markovian” character of the additive stem distance and the structure of a DNA code specified by its invariance under the Watson–Crick transformation. In particular, our lower bound is established with the help of an ensemble of random codes where distribution of independent codewords is defined by a stationary Markov chain.