01672644

j

01672644 Ling, San Sol\'e, Patrick Type II codes over $\bbfF_4+u\bbfF_4$. Eur. J. Comb. 22, No.7, 983-997 (2001). 2001 Elsevier Science (Academic Press), London EN type II codes self-dual codes doubly-even codes Zbl 0947.94023 Zbl 0876.94053

doi:10.1006/eujc.2001.0509

Self-dual codes over $\bbfF_4$ for the Euclidean scalar product and over $\bbfF_2+u\bbfF_2$ 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 [{\it P. Gaborit}, {\it V. Pless}, {\it P. Sol\'e} and {\it O. Atkin}, Type II codes over $\bbfF_4$ (preprint, 1999) and {\it S. T. Dougherty}, {\it P. Gaborit}, {\it M. Harada} and {\it P. Sol\'e}, 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 [{\it C. Bachoc}, J. Comb. Theory, Ser. A 78, 92-119 (1997; Zbl 0876.94053)], they also give constructions of lattices; specifically $\bbfZ$-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.