Hexacode

In coding theory, the hexacode is length 6 linear code of dimension 3 over the Galois field $GF(4)=\{0,1,omega,omega^2\}$ of 4 elements defined by :$H=\{(a,b,c,f(1),f(omega),f(omega^2)\; :\; f(x):=ax^2+bx+c;\; a,b,cin\; GF(4)\}.$Then $H$ contains 45 codewords of weight 4, 18 codewords of weight 6 andthe zero word. The full automorphism group of the hexacode is $3.S\_6$. The hexacode can be used to describe the Miracle Octad Generatorof R. T. Curtis.

References

*cite book | first = John H. | last = Conway | authorlink = John Horton Conway | coauthors = Sloane, Neil J. A. | year = 1998 | title = Sphere Packings, Lattices and Groups | edition = (3rd ed.) | publisher = Springer-Verlag | location = New York | id = ISBN 0-387-98585-9