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By N.J.A. Sloane

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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]. 2, [ 2] Ch. 4. 10) is a special case of a theorem of Perron [ 78] . 14) has been studied by Assmus and Mattsou [59], [60] , Goethals [67], and many others.

Of the ring of polynomials modulo xn + 1. If we multiply a(x) by x we get rc xa(x) 2 ao x + a1 x + ••• + a n-2 x = a n- 1 n-1 + ao x + ••• + a n- 2 x (in IR n ) which is associated with the vector ( an-1' ao x in «Rncorresponds to a cyclic shift of the vector. + a n-1 x n n-1 , ... 2) Definition An ideal~ of «Rnis a linear subspace of IRn}, .

3 ), the distance between any two vectors of c' is even. The minimum distance cannot be less than d (which is odd), so must be d + 1. D. 5) we obtain the (8,16,4) extended Hamming code. 8. 4 The [ 24,12,8] Extended Golay Code and Its Properties The Golay code is probably the most important of all codes. In this section we give an elementary definition of the code, and establish a nurober of its properties. 1) Definition The extended Golay code t:§ has generator matrix: +------- G= ,Q ------;~+------- 00 0 1 2 3 4 5 6 7 8 9 10 l 1 1 1 1 1 1 1 1 1 1 1 00 0 1 2 3 4 5 6 7 8 9 10 1 1 1 r ----- 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 (The co1umns are 1abelled ~ ..

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