In DNA data storage and computation, DNA strands are required to meet certain combinatorial constraints. This paper shows how some of these constraints can be achieved simultaneously. First, we use the algebraic structure of irreducible cyclic codes over finite fields to generate cyclic DNA codes that satisfy reverse and complement properties. We show how such DNA codes can meet constant guanine-cytosine content constraint by MacWilliams-Seery algorithm. Second, we consider fulfilling the run-length constraint in parallel with the above constraints, which allows a maximum predetermined number of consecutive duplicates of the same symbol in each DNA strand. Since irreducible cyclic codes can be represented in terms of the trace function over finite field extensions, the linearity of the trace function is used to fulfill a predefined run-length constraint. Thus, we provide an algorithm for constructing cyclic DNA codes with the above properties including run-length constraint. We show numerical examples to demonstrate our algorithms generating such a set of DNA strands with all the prescribed constraints.

We prove that binary images of irreducible cyclic codes C over GF(2m) and binary concatenated codes of C and a binary [m+1,m,2] even-parity code are optimal (in the sense that they meet the Griesmer bound with equality) and proper, if a root of the check polynomial of C is primitive over GF(2m) or its extensions.