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In [[coding theory]], the '''hexacode''' is a length 6 [[linear code]] of dimension 3 over the [[Galois field]] <math>GF(4)=\{0,1,\omega,\omega^2\}</math> of 4 elements defined by 
:<math>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>GF(4)</math>.
Then <math>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>3.S_6</math>. The hexacode can be used to describe the [[Miracle Octad Generator]]
of R. T. Curtis.

==References==

*{{cite book | first = John H. | last = Conway | authorlink = John Horton Conway | coauthors  = [[Neil Sloane|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}}

[[Category:Coding theory]]