Designs, Codes and Cryptography

, Volume 75, Issue 1, pp 81–95 | Cite as

Perfect codes in the discrete simplex



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.


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

Mathematics Subject Classification

94B25 05B40 52C17 05C12 68R99 


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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Electrical EngineeringUniversity of Novi SadNovi SadSerbia

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