Polar Codes with Balanced Codewords

The imbalance of a binary word refers to the absolute difference between the number of ones and zeros in the word. Motivated by applications in DNA-based data storage and the success of polar codes, we study the problem of reducing imbalance in the codewords of a polar code. To this end, we adapt the technique of Mazumdar, Roth, and Vontobel by considering balancing sets that correspond to low-order Reed-Muller (RM) codes. Such balancing sets are likely to be included as subcodes in polar codes.Specifically, using the first-order RM code, we show that any message can be encoded into a length-n polar codeword with imbalance at most o(n) in O(nlogn)-time. We then reduce the imbalance even further using two methods. First, we constrain the ambient space $\mathbb{X}$ and analyze the imbalance that the first-order RM code can achieve for words in $\mathbb{X}$. We demonstrate that for codelengths up to 128, the first-order RM code achieves zero imbalance for appropriate choices of $\mathbb{X}$ that sacrifice only a few message bits. Second, we augment the balancing set by considering higher order RM codes. We give a simple recursive upper bound for the guaranteed imbalance of RM codes. We also prove that the second-order RM code $\mathbb{R}\mathbb{M}\left( {2,m} \right)$ balances all even-weight words for m ⩽ 5, while the RM code of order m − 3 balances all even-weight words for m ⩾ 5.