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Perfect codes in the discrete simplex

Abstract

We study the problem of existence of (nontrivial) perfect codes in the discrete \( n \)-simplex \( \Delta _{\ell }^n := \left\{ \left( \begin{array}{l} x_0, \ldots , x_n \end{array}\right) : x_i \in {\mathbb {Z}}_{+}, \sum _i x_i = \ell \right\} \) 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 \( \Delta _{\ell }^1 \) exist for any \( \ell \ge 2e + 1 \), the 2-simplex \( \Delta _{\ell }^2\) 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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Notes

  1. 1.

    In the graph theoretic literature, 1-perfect codes are also known as efficient dominating sets (see, e.g., [4]).

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Acknowledgments

The authors are very grateful to the reviewers for a detailed reading and many useful comments on the original version of the manuscript. This work was supported by the Ministry of Science and Technological Development of the Republic of Serbia (Grants TR32040 and III44003). Part of the work was done while M. Kovačević was visiting Aalborg University, Denmark, under the support of the COST action IC1104. He is very grateful to the Department of Electronic Systems, and in particular to Čedomir Stefanović and Petar Popovski, for their hospitality.

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Correspondence to Mladen Kovačević.

Additional information

Communicated by P. Charpin.

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Kovačević, M., Vukobratović, D. Perfect codes in the discrete simplex. Des. Codes Cryptogr. 75, 81–95 (2015). https://doi.org/10.1007/s10623-013-9893-5

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Keywords

  • Multiset codes
  • Permutation channel
  • Discrete simplex
  • Perfect codes
  • Sphere packing
  • Integer codes
  • Manhattan metric

Mathematics Subject Classification

  • 94B25
  • 05B40
  • 52C17
  • 05C12
  • 68R99