Homopolymer-Free Constant Weighted Sum Sequences: Code Rate and Applications to DNA Codes

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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathbf{w}$</tex> 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.