离散单纯形中的完全码
Perfect Codes in the Discrete Simplex
摘要：
研究了离散$ n $ -单纯形$ Δ_{\ell}^n := \left\{ \begin{pmatrix} x_0, \ldots, x_n \end{pmatrix} : x_i \in \mathbb{Z}_{+}, \sum_i x_i = \ell \right\} $在$ \ell_1 $度量下的（非平凡）完美码的存在性问题.这个问题的动机是由所谓的多集码，这是最近推出的作者作为适当的结构，在置换通道的纠错。证明了对于任何$ \ell \geq 2e + 1 $，$ 1 $ -单纯形$ Δ_{\ell}^1 $中的$ e $ -完美码都存在，$ 2 $ -单纯形$ Δ_{\ell}^2 $允许$ e $ -完美码当且仅当$ \ell = 3e + 1 $，而在高维单纯形中不存在完美码.换句话说，完美多集码只存在于二进制和三进制字母表上。
英文摘要：
We study the problem of existence of (nontrivial) perfect codes in the discrete $ n $-simplex $ Δ_{\ell}^n := \left\{ \begin{pmatrix} x_0, \ldots, x_n \end{pmatrix} : x_i \in \mathbb{Z}_{+}, \sum_i x_i = \ell \right\} $ under $ \ell_1 $ metric. The problem is motivated by the so-called multiset codes, which have recently been introduced by the authors as appropriate constructs for error correction in the permutation channels. It is shown that $ e $-perfect codes in the $ 1 $-simplex $ Δ_{\ell}^1 $ exist for any $ \ell \geq 2e + 1 $, the $ 2 $-simplex $ Δ_{\ell}^2 $ admits an $ e $-perfect code if and only if $ \ell = 3e + 1 $, while there are no perfect codes in higher-dimensional simplices. In other words, perfect multiset codes exist only over binary and ternary alphabets.
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