Type II codes over \(\mathbb{F}_4+u\mathbb{F}_4\) (Q5949035)

Self-dual codes over \(\mathbb{F}_4\) for the Euclidean scalar product and over \(\mathbb{F}_2+u\mathbb{F}_2\) have received some attention lately. In the present article, codes over an alphabet \(R\) of size 16 that contains both alphabets as subrings are studied. As a consequence, the authors obtain via suitable Gray maps self-dual codes over these two subrings, which, in turn, give self-dual binary codes by the Gray maps of [\textit{P. Gaborit}, \textit{V. Pless}, \textit{P. Solé} and \textit{O. Atkin}, Type II codes over \(\mathbb{F}_4\) (preprint, 1999) and \textit{S. T. Dougherty}, \textit{P. Gaborit}, \textit{M. Harada} and \textit{P. Solé}, IEEE Trans. Inf. Theory 45, 32-45 (1999; Zbl 0947.94023)]. In particular, they introduce a subclass (Type II codes) of these self-dual codes over \(R\) that yield, after double Gray mapping, doubly-even binary codes. Following a trend illustrated in [\textit{C. Bachoc}, J. Comb. Theory, Ser. A 78, 92-119 (1997; Zbl 0876.94053)], they also give constructions of lattices; specifically \(\mathbb{Z}\)-lattices, Gaussian lattices and lattices over the golden integers via Construction A. A connection with Tits' quaternionic construction of the Leech lattice is pointed out.