Hexacode

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

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

It is a 3-dimensional subspace of the vector space of dimension 6 over [math]\displaystyle{ GF(4) }[/math]. Then [math]\displaystyle{ H }[/math] contains 45 codewords of weight 4, 18 codewords of weight 6 and the zero word. The full automorphism group of the hexacode is [math]\displaystyle{ 3.S_6 }[/math]. 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.) ed.). New York: Springer-Verlag. ISBN 0-387-98585-9. https://archive.org/details/spherepackingsla0000conw_b8u0.