User rishig - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T21:14:15Z http://mathoverflow.net/feeds/user/27235 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/109635/automorphisms-of-subgroup-of-hamming-cube-under-distance-constraint Automorphisms of subgroup of hamming cube under distance constraint rishig 2012-10-14T18:27:27Z 2013-05-24T17:22:00Z <p>Let \$S\$ be a subset of <code>\$\{0,1\}^n\$</code> such that any two elements of \$S\$ are at least (Hamming) distance 5 apart. I'm looking for an upper bound on the size of the automorphism group of \$S\$.</p> <p>There's a trivial upper bound of \$2^nn!\$ (the number of automorphisms of <code>\$\{0,1\}^n\$</code>), and an easy lower bound of \$2^{n/5}(n/5)!\$ (take S to be all elements of the form \$xxxxx\$, where \$x\$ is a bitstring of length \$n/5\$). </p> <p>Any bound of the form \$n!/n^{cn}\$ for any \$c>0\$ would be helpful.</p> http://mathoverflow.net/questions/109635/automorphisms-of-subgroup-of-hamming-cube-under-distance-constraint Comment by rishig rishig 2012-10-19T06:23:06Z 2012-10-19T06:23:06Z Thanks for the comment. Haven't found anything yet that I've gotten to work, but I'll post here if I do. http://mathoverflow.net/questions/109635/automorphisms-of-subgroup-of-hamming-cube-under-distance-constraint/109898#109898 Comment by rishig rishig 2012-10-19T04:44:51Z 2012-10-19T04:44:51Z Thanks for your response. I'm referring to automorphisms of the hypercube [1], which are substantially more restricted than \$F_2^n\$. For instance the linear map [[1 1] [1 0]] is an automorphism of \$F_2^2\$ but not of the square. [1] <a href="http://en.wikipedia.org/wiki/Hyperoctahedral_group" rel="nofollow">en.wikipedia.org/wiki/Hyperoctahedral_group</a>