By Scott R.F.
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Additional info for A treatise on the theory of determinants
By Hensel-lifting this polynomial (using say Graeffe's root-squaring method, cf. [HaKCSS], p. 307) to &:: 4 we obtain g4 (x) = x 11 + 2x 10 - x9 - x7 - x6 - x 5 + 2x 4 +x-I , a divisor of x 23 -I (mod 4). By appending a zero-sum check symbol to the cyclic code generated by g 4 (x), we obtain a self-dual code of length 24 over &:: 4 • Applying Construction A4 (cf. Chapter 5), that is, taking all vectors in &::24 which when read mod 4 are in the code, we obtain the Leech lattice. 759 of shape 116 08 and 48 of shape 4 1023 .
Thus the coordination sequence of E~ begins I, 240, 9120, 121680, 864960, . .. For further examples see [BattVe98], [CSLDL 7], [GrBS], and [SloEIS]. We do not know the coordination sequence of the Leech lattice. In [CSLDL 7] we also show that among all the Barlow packings in three dimensions (those obtained by stacking A 2 layers, cf. [CoS195a]) the hexagonal close packing has the greatest coordination sequence, and the face-centered cubic lattice the smallest. On 2 +2 ::=:: S(n) ::=:: [21n 2 /2] +2 (n > 0) .
In [CoS191]] we prove all cases except n [Smi91]) of the following proposition: = 2 (handled by Smith The lattices with minimal Erdos number are (up to a scale factor) the even lattices of minimal determinant. For n = 0, 1, 2, . . these determinants are 1, 2, 3, 4, 4, 4, 3, 2, 1, 2, 3, 4, 4, 4, this sequence continuing with period 8. 2 For n ::5 2 these definitions must be modified. For n = 0 and 1 we set E = 1, while for n = 2 we define F by F = limx... ogx. 5581, (rounded to 4 decimal places), while for each n ~ 11 there are two or more such lattices.