Hexacode
In coding theory, the hexacode is a length 6 linear code of dimension 3 over the Galois field ![GF(4)=\{0,1,\omega,\omega^2\}](/Images/godic/202501/18/75f18d2754e103def0f343aa452b681d4437.png")
of 4 elements defined by
It is a 3-dimensional subspace of the vector space of dimension 6 over
![GF(4)](/Images/godic/202501/18/990fa5cc1dfcea5d204aab52442f10994437.png")
.
Then ![H](/Images/godic/202501/18/c1d9f50f86825a1a2302ec2449c171964437.png")
contains 45 codewords of weight 4, 18 codewords of weight 6 and
the zero word. The full automorphism group of the hexacode is
![3.S_6](/Images/godic/202501/18/c479cdd0517ce80d9f183a924dc1199b4437.png")
. The hexacode can be used to describe the Miracle Octad Generator
of R. T. Curtis.