Optimal Almost-Balanced Sequences

This paper presents a novel approach to address the constrained coding challenge of generating almost-balanced sequences. While strictly balanced sequences have been well studied in the past, the problem of designing efficient algorithms with small redundancy, preferably constant or even a single bit, for almost balanced sequences has remained unsolved. A sequence is ϵ(n)-almost balanced if its Hamming weight is between 0.5n±ϵ(n). It is known that for any algorithm with a constant number of bits, ϵ(n) has to be in the order of Θ(√n), with O(n) average time complexity. However, prior solutions with a single redundancy bit required ϵ(n) to be a linear shift from n/2. Employing an iterative method and arithmetic coding, our emphasis lies in constructing almost balanced codes with a single redundancy bit. Notably, our method surpasses previous approaches by achieving the optimal balanced order of Θ(√n). Additionally, we extend our method to the non-binary case, considering q-ary almost polarity-balanced sequences for even q, and almost symbol-balanced for q=4. Our work marks the first asymptotically optimal solutions for almost-balanced sequences, for both, binary and non-binary alphabet.