Hexacode

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

H = { ( a , b , c , f ( 1 ) , f ( ω ) , f ( ω 2 ) ) : f ( x ) := a x 2 + b x + c ; a , b , c ∈ G F ( 4 ) } . {\displaystyle H=\{(a,b,c,f(1),f(\omega ),f(\omega ^{2})):f(x):=ax^{2}+bx+c;a,b,c\in GF(4)\}.}

It is a 3-dimensional subspace of the vector space of dimension 6 over G F ( 4 ) {\displaystyle GF(4)} . Then H {\displaystyle 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. S 6 {\displaystyle 3.S_{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.) ed.). New York: Springer-Verlag. ISBN 0-387-98585-9.