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
$\varepsilon(n)$
-almost balanced if its Hamming weight is between
$0.5n\pm\varepsilon(n)$
. It is known that for any algorithm with a constant number of bits,
$\varepsilon(n)$
has to be in the order of
$\Theta(\sqrt{n})$
, with
$\mathcal{O}(n)$
average time complexity. However, prior solutions with a single redundancy bit required
$\varepsilon(n)$
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
$\Theta(\sqrt{n})$
. 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.