Dense Sphere Packings from New Codes

The idea behind the coset code construction (see [G.D. Forney, Coset Codes, IEEE Transactions on Information Theory, Part I: Introduction and Geometrical Classification, pp. 1123–1151; Part II: Binary lattices and related codes, pp. 1152–1187; F.R. Kschischang and S. Pasupathy, IEEE Transactions on Information Theory 38 (1992), 227–246.]) is to reduce the construction of sphere packings to error-correcting codes in a unified way. We give here a short self-contained description of this method. In recent papers [J. Bierbrauer and Y. Edel, IEEE Transactions on Information Theory 43 (1997), 953–968; J. Bierbrauer and Y. Edel, Finite Fields and Their Applications 3 (1997), 314–333; J. Bierbrauer and Y. Edel, IEEE Transactions on Information Theory 44 (1998), 1993; J. Bierbrauer, Y. Edel, and L. Tolhuizen, Finite Fields and Their Applications, submitted for publication.] we constructed a large number of new binary, ternary and quaternary linear error-correcting codes. In a number of dimensions our new codes yield improvements. Recently Vardy [A. Vardy, Inventiones Mathematicae 121, 119–134; A. Vardy, Density doubling, double-circulants, and new sphere packings, Trans. Amer. Math. Soc. 351 (1999), 271–283.] has found a construction, which yields record densities in dimensions 20, 27, 28, 29 and 30. We give a short description of his method using the language of coset codes. Moreover we are able to apply this method in dimension 18 as well, producing a sphere packing with a record center density of (3/4)9.