Optimal Almost-Balanced Sequences
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866909201850171392 |
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| author | Bar-Lev, Daniella Kobovich, Adir Leitersdorf, Orian Yaakobi, Eitan |
| author_facet | Bar-Lev, Daniella Kobovich, Adir Leitersdorf, Orian Yaakobi, Eitan |
| contents | 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 $Θ(\sqrt{n})$, with $O(n)$ 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 $Θ(\sqrt{n})$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_08625 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Optimal Almost-Balanced Sequences Bar-Lev, Daniella Kobovich, Adir Leitersdorf, Orian Yaakobi, Eitan Information Theory 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 $Θ(\sqrt{n})$, with $O(n)$ 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 $Θ(\sqrt{n})$. 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. |
| title | Optimal Almost-Balanced Sequences |
| topic | Information Theory |
| url | https://arxiv.org/abs/2405.08625 |