For any alphabet, consider sequences that both contain no consecutive repeated symbols and have the same number of occurrences of symbols from a specified subset of the alphabet. These properties are particularly valuable in applications such as DNA data storage, where avoiding repeated patterns improves stability and robustness. A Homopolymer-Free Constant Weighted Sum (HFCWS) code, defined over an alphabet, is a set of sequences that satisfies two specific properties: (1) Constant Weighted Sum (CWS) Property: For a designated subset of the alphabet, the number of total occurrences of symbols from a given subset of the alphabet remains the same across all sequences in the code. (2) Homopolymer-Free (HF) Property: Each sequence in the code is Homopolymer-Free, with no consecutive identical symbols. In this paper, for a given code defined over an alphabet, we enumerate the total number of sequences that satisfy the Constant Weighted Sum property with a specific weight$\mathbf{w}$and the Homopolymer-Free property. The asymptotic code rate of HFCWS codes is derived for both large and small constant weighted sums. Also, a lower bound on asymptotic code rate is established for the case when the constant weighted sum is approximately half of the sequence length. We establish an equivalence between HFCWS codes over the quaternary alphabet and Homopolymer-Free GC-Balanced DNA codes, demonstrating that our derived code rate formula not only aligns with existing results but also provides new theoretical insights for Homopolymer-Free GC-Balanced DNA codes.

Inspired by G. Tenengolts' 1984 construction of nonbinary single error-correcting codes for deletion and insertion errors, we develop families of DNA codes that simultaneously address multiple biological and combinatorial constraints. These codes satisfy Hamming, Reverse, Reverse-Complement, and GCcontent constraints while maintaining single error-correction capabilities for insertion, deletion, and substitution errors. Additionally, our constructed codes are free from homopolymers exceeding run length four and from secondary structures with stem lengths greater than two, both in individual codewords and their concatenations. This work presents the first known construction of DNA codes capable of single-error correction (insertion, deletion, and substitution) while satisfying multiple constraints: the GC-content constraint, Hamming constraint, Reverse constraint, and Reverse-Complement constraint. Additionally, these codes are constructed to be free from homopolymers exceeding run length four and secondary structures with stem lengths greater than two.

For any given alphabet of size q, a Homopolymer Free code (HF code) refers to an (n, M, d)qcode of length n, size M and minimum Hamming distance d, where all the codewords are homopolymer free sequences. For any given alphabet, this work provides upper and lower bounds on the maximum size of any HF code using Sphere Packing bound and Gilbert-Varshamov bound. Further, upper and lower bounds on the maximum size of HF codes for various HF code families are calculated. Also, as a specific case, upper and lower bounds are obtained on the maximum size of homopolymer free DNA codes.

Motivated from Reed-Muller codes, families of reversible, reversible-complement, and GC-balanced DNA codes are constructed from the ring ℤ6. In addition, DNA codewords are free from any secondary structures having stems of length more than two and homopolymers with run-length more than three. Also, DNA strings obtained from concatenations of those DNA codewords avoid secondary structures and homopolymers. We have also given a lower bound on the size of DNA codes with all these properties together.

In this paper, we present a novel design strategy of DNA codes with length 3n over the non-chain ring ℤ4+uℤ4+u2ℤ4with 64 elements and u3=1, where n denotes the length of a code over R. We first study and analyze a distance conserving map defined over the ring R into the length-3 DNA sequences. Then, we derive some conditions on the generator matrix of a linear code over R, which leads to a DNA code with reversible, reversible-complement, homopolymer 2-run-length, and3wn-GC-content constraints for integer w (0 ≤ w ≤ 3n). Finally, we propose a new construction of DNA codes using Reed-Muller type generator matrices. This allows us to obtain DNA codes with reversible, reversible-complement, homopolymer 2-run-length, and -GC-content constraints.

In this work, we study the DNA codes arising from the ring R=cZ4+wZ4, where w2=2+2w with 16 elements. We establish a one to one correspondence, between the elements of the ring R and all the DNA words of length 2, by defining a distance preserving Gau map φ. Using this map, we give several new classes of DNA codes which are reversible and reversible-complement. We also give general conditions on the generator matrix of a code over R which produces DNA codes with the required properties using the Gau map. Some of the constructed DNA codes are optimal.