Perfect codes in the discrete simplex

We study the problem of existence of (nontrivial) perfect codes in the discrete n -simplex Δ ℓ n : = x 0 , … , x n : x i ∈ Z + , ∑ i x i = ℓ under ℓ 1 metric. The problem is motivated by the so-called multiset codes, which have recently been introduced by the authors as appropriate constructs for er...

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Bibliographic Details
Published in:Designs, codes, and cryptography Vol. 75; no. 1; pp. 81 - 95
Main Authors: Kovačević, Mladen, Vukobratović, Dejan
Format: Journal Article
Language:English
Published: Boston Springer US 01.04.2015
Subjects:
Circuits
Coding and Information Theory
Computer Science
Cryptology
Data Structures and Information Theory
Discrete Mathematics in Computer Science
Information and Communication
Integer codes
94B25
05C12
68R99
Discrete simplex
05B40
Sphere packing
Multiset codes
Manhattan metric
52C17
Permutation channel
Perfect codes
ISSN:0925-1022, 1573-7586
Online Access:Get full text
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Summary:We study the problem of existence of (nontrivial) perfect codes in the discrete n -simplex Δ ℓ n : = x 0 , … , x n : x i ∈ Z + , ∑ i x i = ℓ under ℓ 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 ℓ 1 exist for any ℓ ≥ 2 e + 1 , the 2-simplex Δ ℓ 2 admits an e -perfect code if and only if ℓ = 3 e + 1 , while there are no perfect codes in higher-dimensional simplices. In other words, perfect multiset codes exist only over binary and ternary alphabets.
ISSN:0925-1022
1573-7586
DOI:10.1007/s10623-013-9893-5