Abstract
We consider the construction of group block codes, i.e., subgroups of Gn, the n-fold direct product of a group G. Two concepts are introduced that make this construction similar to that of codes over gf(2). The first concept is that of an indecomposable code. The second is that of a parity-check matrix. As a result, group block codes over a decomposable Abelian group of exponent dm can be seen as block codes over the ring of residues modulo dm, and their minimum Hamming distance can be easily determined. We also prove that, under certain technical conditions, (n, k) systematic group block codes over non-Abelian groups are asymptotically bad, in the sense that their minimum Hamming distance cannot exceed [n/k].
Résumé
On considère la construction de codes en bloc sur des groupes, c’est-à-dire de sous-groupes de Gn, le produit direct n fois ďun groupe G par lui-même. Deux concepts introduits rendent cette construction similaire à celle des codes sur gf(2). Le premier est celui ďun code indécomposable. Le second et celui ďune matrice de vérification de parité. Il en résulte que des codes en bloc sur un groupe abélien décomposable ďexposant dm peuvent être interprétés comme des codes en bloc sur ľanneau des résidus modulo dm, et leur distance de Hamming minimale peut aisément être déterminée. On montre aussi que, sous certaines conditions techniques, les codes en bloc (n, k) systématiques sur des groupes non abéliens sont asymptotiquement mauvais, en ce sens que leur distance de Hamming minimale ne peut pas dépasser [n/k].
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This research was sponsored by the Italian National Research Council (cnr) under Progetto Finalizzato Trasporti.
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Biglieri, E., Elia, M. On the construction of group block codes. Ann. Télécommun. 50, 817–823 (1995). https://doi.org/10.1007/BF02997785
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DOI: https://doi.org/10.1007/BF02997785
Key words
- Error correcting code
- Block code
- Linear code
- Finite group
- Abelian group
- Hamming distance
- Minimal distance
- Group theory