Improved Upper Bound for the Size of a Trifferent Code
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911770845642752 |
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| author | Bhandari, Siddharth Khetan, Abhishek |
| author_facet | Bhandari, Siddharth Khetan, Abhishek |
| contents | A subset $\mathcal{C}\subseteq\{0,1,2\}^n$ is said to be a $\textit{trifferent}$ code (of block length $n$) if for every three distinct codewords $x,y, z \in \mathcal{C}$, there is a coordinate $i\in \{1,2,\ldots,n\}$ where they all differ, that is, $\{x(i),y(i),z(i)\}$ is same as $\{0,1,2\}$. Let $T(n)$ denote the size of the largest trifferent code of block length $n$. Understanding the asymptotic behavior of $T(n)$ is closely related to determining the zero-error capacity of the $(3/2)$-channel defined by Elias'88, and is a long-standing open problem in the area. Elias had shown that $T(n)\leq 2\times (3/2)^n$ and prior to our work the best upper bound was $T(n)\leq 0.6937 \times (3/2)^n$ due to Kurz'23. We improve this bound to $T(n)\leq c \times n^{-2/5}\times (3/2)^n$ where $c$ is an absolute constant. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_02390 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Improved Upper Bound for the Size of a Trifferent Code Bhandari, Siddharth Khetan, Abhishek Information Theory Discrete Mathematics Combinatorics A subset $\mathcal{C}\subseteq\{0,1,2\}^n$ is said to be a $\textit{trifferent}$ code (of block length $n$) if for every three distinct codewords $x,y, z \in \mathcal{C}$, there is a coordinate $i\in \{1,2,\ldots,n\}$ where they all differ, that is, $\{x(i),y(i),z(i)\}$ is same as $\{0,1,2\}$. Let $T(n)$ denote the size of the largest trifferent code of block length $n$. Understanding the asymptotic behavior of $T(n)$ is closely related to determining the zero-error capacity of the $(3/2)$-channel defined by Elias'88, and is a long-standing open problem in the area. Elias had shown that $T(n)\leq 2\times (3/2)^n$ and prior to our work the best upper bound was $T(n)\leq 0.6937 \times (3/2)^n$ due to Kurz'23. We improve this bound to $T(n)\leq c \times n^{-2/5}\times (3/2)^n$ where $c$ is an absolute constant. |
| title | Improved Upper Bound for the Size of a Trifferent Code |
| topic | Information Theory Discrete Mathematics Combinatorics |
| url | https://arxiv.org/abs/2402.02390 |