Hexacode

In coding theory, the hexacode is a length 6 linear code of dimension 3 over the Galois field ${\displaystyle GF(4)=\{0,1,\omega ,\omega ^{2}\}}$ of 4 elements defined by

${\displaystyle H=\{(a,b,c,f(1),f(\omega ),f(\omega ^{2})):f(x):=ax^{2}+bx+c;a,b,c\in GF(4)\}.}$

It is a 3-dimensional subspace of the vector space of dimension 6 over ${\displaystyle GF(4)}$. Then ${\displaystyle H}$ contains 45 codewords of weight 4, 18 codewords of weight 6 and the zero word. The full automorphism group of the hexacode is ${\displaystyle 3.A_{6}}$.¹ The hexacode can be used to describe the Miracle Octad Generator of R. T. Curtis.