The set of all binary vectors with 12 components forms a field with 2^12 elements containing 000000000000 and another 65\*63 elements. Is it possible to partition these elements into 65 subgroups of 63 vectors so that each of them is closed under binary addition (XOR)?


For example, if we had 4 components instead of 12, the field contains 0000 and another 3\*5 elements, we can partition them as follows:


0000


0001
0010
0011


0100
1000
1100


0101
1010
1111


0110
1011
1101


0111
1001
1110


so that, whenever we add two elements from the same group, we get another from the same group. We have five subgroups closed under XOR that only share the element 0000. Could we do the same with vectors with 12 components? Is there some theorem that governs this property for more components?


Thank you!



Binary algebra, is it possible to partition the elements in GF(2^12) into 65 subgroups closed under addition?
$\mathrm{GF}(2^{12})$ is a two-dimensional vector space over $\mathrm{GF}(2^6)$. The one-dimensional subspaces are all disjoint (barring the zero vector), and contain 63 nonzero elements each.


