Ternary code construction of unimodular lattices and self-dual codes over \(\mathbb Z_6\)
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.
scientific article; zbMATH DE number 1879617
scientific article