Perfect codes in the discrete simplex
Perfect codes in the discrete simplex
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DOI：
10.1007/s10623-013-9893-5
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发表时间：
2013-07
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影响因子：
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通讯作者：
Mladen Kovačević;D. Vukobratović
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文献类型：
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作者：
Mladen Kovačević;D. Vukobratović
研究了离散单形$$\Delta_{\ell}^n：=\Left\Left(\Begin{array{L}x0，\ldots，xn\end{array}\right)：xi\in{mathbb{Z}}{+}，\sum_ixi=\ell\right$下的(非平凡)完全码的存在性问题。该问题是由所谓的多集码引起的，该多集码是作者最近引入的用于置换信道中纠错的适当构造。证明了1-单形中存在-完美码，2-单形允许一个-完美码当且仅当，而高维单形中不存在完美码。换句话说，完美多集码只存在于二进制和三进制字母表上。
We study the problem of existence of (nontrivial) perfect codes in the discrete-simplex $$ \Delta _{\ell }^n := \left\{ \left( \begin{array}{l} x_0, \ldots , x_n \end{array}\right) : x_i \in {\mathbb {Z}}_{+}, \sum _i x_i = \ell \right\} $$ undermetric. 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-perfect codes in the 1-simplexexist for any, the 2-simplexadmits an-perfect code if and only if, while there are no perfect codes in higher-dimensional simplices. In other words, perfect multiset codes exist only over binary and ternary alphabets.