Ternary code construction of unimodular lattices and self-dual codes over \(\mathbb Z_6\)
type: Item
instance of: scholarly article
title: Ternary code construction of unimodular lattices and self-dual codes over \(\mathbb Z_6\)
zbMATH Open document ID: 1027.94022
DOI: 10.1023/A:1021185314365
author: Masaaki Harada
author: Masaaki Kitazume
author: Michio Ozeki
published in: Journal of Algebraic Combinatorics
publication date: 2003-03-11T00:00:00Z
review text: After a brief introduction to the known relationships between codes and lattices, construction A and a ternary code construction given in [\textit{M. Ozeki}, Ternary code construction of even unimodular lattices, Theorie des nombres, C. R. Conf. Int., Quebec/Can. 1987 (de Gruyter, Berlin), 772-784 (1989; Zbl 0686.10025)] are considered. An error in that paper involving the minimum weights when the dimension is divisible by 12 is corrected. The ternary code construction is reconsidered here using shadow lattices. The reader who is unfamiliar with shadow theory may wish to consult additional literature [e.g. \textit{J. H. Conway} and \textit{N. J. A. Sloane}, J. Number Theory 72, 357-362 (1998; Zbl 0917.11027) or \textit{E. Rains} and \textit{N. J. A. Sloane}, ibid. 73, 359-389 (1998; Zbl 0917.11026)], as some of the notation, hence the definition of the shadow lattice, is not clearly explained here. The methods are applied to give explicit extremal Type II \(\mathbb Z_6\)-codes of lengths 48, 56, and 64 as well as extremal Type I \(\mathbb Z_6\)-codes of lengths 44, 60, and 68.
Mathematics Subject Classification ID: 94B05
Mathematics Subject Classification ID: 11H71
zbMATH DE Number: 1879617
zbMATH Keywords: extremal self-dual codes
zbMATH Keywords: extremal unimodular lattices
zbMATH Keywords: shadows
zbMATH Keywords: ternary codes
reviewed by: Katherine Roegner
MaRDI profile type: Publication
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full work available at URL: a:1021185314365
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description: scientific article; zbMATH DE number 1879617