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Powell and G. High man, Academic Press, N. Y. -M. (1971), On the Golay Perfeet Binary Code, JCT 11:178-186. [ 68 ] Goethals, J. -M. and S. L. Snover (1972 ), Nearly Perfeet Binary Codes, DM 3:65-88. [ 69 ] Karlin, M. (1969), New Binary Coding Results by Circulants, PGIT 15:81-92. References 43 [ 70 ] Karlin, M. J. MacWilliams (197 ), An Elementary Construction of The Golay Code, to appear. [71 ] Leech, J. (1964), Some Sphere Packings in Higher Space, CJM 16:657-682. [ 72 Leech, J. (1967), Notes on Sphere Packings, CJM 19:251-267.

1 0. In 1967 Leech [ 72 ] used the Golay code to construct a very dense sphere packing in 24-dimensional Euclidean space. Conway [ 63]- [66] has extensively studied the symmetry group of this packing. Many other sphere packings based on codes have been given in [73]- [75]. For example we constructed a nonlattice sphere packing in 1 0-dimensional space which is denser than what is thought to be the densest lattice packing. 2) and the connection between the Golay code and the Mathieu group M24 , see Assmus and Mattsou [59], Berlekamp [61], and Conway [ 66].

N-1}. Showthatf(x) has coefficients which are 0 and 1 if and only if ke K => 2ke K modulo n. 10) The basic inversion formula The vector ~ = (ao ,a1 , ... ,a0 _ 1) may be· recovered from the associated polynomial a(x)= a 0 + a 1 x + ••• +a0 _1x 0 -l by § 5. 7 The BCH Bound This is a very useful lower bound on the minimum distance of a cyclic code. As usual o: denotes a primitive n-th root of unity. Then the minimum distance of "€ is at least d0 • Proof. For any codeward c(x) eCC we have c ( ol) = 0, ViEK:.

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