Designs, Codes and Cryptography

, Volume 75, Issue 1, pp 81–95

Perfect codes in the discrete simplex

Article

DOI: 10.1007/s10623-013-9893-5

Cite this article as:
Kovačević, M. & Vukobratović, D. Des. Codes Cryptogr. (2015) 75: 81. doi:10.1007/s10623-013-9893-5

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.

Keywords

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

Mathematics Subject Classification

94B25 05B40 52C17 05C12 68R99 

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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