The cubed root symbol $\omega$ may be thought of as a non zero element in the four element field. In the hexacode description of the miracle octad there are $3 \times 6$ arrays, that is, three length $6$ words in four letters. A key three word matrix is built from the $3 \times 3$ identity and the Fourier operator,
in analogy to the (gauge fixed) $2 \times 4$ arrays that create a Grassmannian Minkowski space in the $2 \times 2$ twistor formalism. Conway has a similar construction for the ternary code, known as miniMOG. The hexacode is used to specify five points from which a Steiner octad is created.
6 years ago