Hexacode

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

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

It is a 3-dimensional subspace of the vector space of dimension 6 over $GF(4)$. Then $H$ contains 45 codewords of weight 4, 18 codewords of weight 6 and the zero word. The full automorphism group of the hexacode is $3.A_6$. The hexacode can be used to describe the Miracle Octad Generator of R. T. Curtis.

References

• Conway, John H.; Sloane, Neil J. A. (1998). Sphere Packings, Lattices and Groups (3rd ed.). New York: Springer-Verlag. ISBN 0-387-98585-9. https://archive.org/details/spherepackingsla0000conw_b8u0.