Wikipedia:Reference desk/Archives/Mathematics/2019 December 4

= December 4 =

number of corners + one offs?
For an N-cube, what is the maximum number of non overlapping sets where each set consists of a specific corner and the corners one away from it? So for N=2, the answer is 1, since that takes up three of the corners, for N=3, the answer is 2, and combined they cover everything. For N=4, only two can be put in, even though 3 would cover 15 of the 16 corners, they can't all be put in. Any idea how to calculate the number for any given N?Naraht (talk) 14:37, 4 December 2019 (UTC)
 * I do not have an answer, but each set contains N+1 vertices and there are 2N vertices in total, so the upper bounding is $$\frac {2^N}{N+1}$$. --CiaPan (talk) 20:19, 4 December 2019 (UTC)
 * The relevant language to look for what's known here is coding theory, particularly error-correcting codes: you're asking for the maximum size of a binary code with minimum Hamming distance 3. The bound given by CiaPan is the Hamming bound (in the case q = 2, d = 3). --JBL (talk) 20:41, 4 December 2019 (UTC)
 * The answer to this particular question is A005864, whose known terms (beginning with N = 1) are 1, 1, 2, 2, 4, 8, 16, 20, 40, 72, 144, 256, 512, 1024, and 2048. (See chapter 9 of J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups" -- I was able to view it on google books but I don't know if that will be true for everyone.) In particular, the answer is not known for N > 15. --JBL (talk) 23:21, 4 December 2019 (UTC)
 * Thanx. Can't see the book right now, that chapter isn't part of the preview for me. I'll look at the other oeis sources.Naraht (talk) 02:58, 7 December 2019 (UTC)
 * Fixed syntax of U template. --CiaPan (talk) 20:59, 9 December 2019 (UTC)