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%I #13 Sep 29 2019 13:15:14
%S 1,2,8,18,40
%N Maximal number of vectors u_1, u_2, ... in R^n with |u_i| = 1 and |u_i - u_j| >= 1 for i, j distinct, where || is L1-norm.
%C L1 norm of (a,b,c,...) = |a|+|b|+|c|+...
%C A lower bound if n = 2^m >= 4: a(n) >= 2n^2 + Sum_{r=2..[log_2 n]} A(2^r,2^(r-1))*A(n,2^r,2^r); compare the Edel-Rains-Sloane paper and the tables of And and Andw mentioned here - _N. J. A. Sloane_, Oct 12 2001. This gives 40 in 4-D, 256 in 8-D (found also by Blokhuis), 2144 in 16-D, etc.
%H A. E. Brouwer, <a href="http://www.win.tue.nl/~aeb/codes/binary-1.html">Tables of general binary codes</a>
%H A. E. Brouwer, <a href="http://www.win.tue.nl/~aeb/codes/Andw.html">Bounds for binary constant weight codes</a>
%H Y. Edel, E. M. Rains and N. J. A. Sloane, <a href="http://www.combinatorics.org/Volume_5/Abstracts/v5i1r22.html">New record kissing numbers in dimensions 32 to 128</a>, Elect. J. Combin.
%H Y. Edel, E. M. Rains and N. J. A. Sloane, <a href="http://neilsloane.com/doc/edel.txt">New record kissing numbers in dimensions 32 to 128</a>
%e It is easier to take the norm to be 2. For n=2: { +-2 0, 0 +-2, +-1 +-1 }; for n=3: { 200 etc. (6) and 110 etc. (12) }; for n=4: { 2000 etc. (8), 1100 etc. (24), .5 .5 .5 .5 etc. (8) }.
%K nonn,more
%O 0,2
%A Aart Blokhuis (aartb(AT)win.tue.nl), Oct 11 2001
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