NEW SPHERE PACKINGS IN DIMENSIONS 9-15 BY JOHN LEECH AND N. J. A. SLOANE

1. Sphere packings from distance 4 codes. Let an (n, M, d) errorcorrecting code be a set of M binary vectors of length n such that any two vectors differ in at least d places. Given such a code with d = 4 a packing of unit spheres in E may be obtained by the following Construction, x = (xi> • • • , xn) is a center of the packing if and only if x is componentwise congruent modulo 2 to a vector in the code. Let the center density of a packing denote the fraction of space covered by the spheres divided by the content of a unit sphere. I t is easily seen that this packing has center density M2~~ and that the number of spheres touching the sphere with center congruent to a codeword c is equal to 2^ + 16.4(c), where -4(c) is the number of codewords differing from c in exactly four places. The codes used here are nonlinear codes and so the packings obtained are nonlattice packings. Nonlinear single error-correcting (8, 20, 3), (9, 38, 3), (10, 72, 3) and (11, 144, 3) codes have been given by Golay [2] and Julin [3]. By annexing a 0 at the end of codewords of even weight and a 1 at the end of codewords of odd weight they are made into codes with d = 4. The packings P9a, PlOa, Plia, P12a of Table 1 are then obtained by applying the above construction to these codes. The codes are not unique and several inequivalent versions of the packings are possible. I t is worth remarking that these codes have more codewords (i.e., are more densely packed!) than any group code of the same length and minimum distance. They have a simple structure, and have recently been generalized to give single error-correcting codes of all lengths 2Sn<3-2~, m^Z, having more codewords than any comparable group code [4], [5].