Lesson plan
Hexacode
In coding theory, the hexacode is a 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,c\in GF(4)\}. It is a 3-dimensional subspace of the vector space of dimension 6 over GF(4) . Then 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.A_6 . The hexacode can be used to describe the Miracle Octad Generator of R. T. Curtis.
Article summary
In coding theory, the hexacode is a 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,c\in GF(4)\}. It is a 3-dimensional subspace of the vector space of dimension 6 over GF(4) . Then 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.A_6 . The hexacode can be used to describe the Miracle Octad Generator of R. T. Curtis.
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Discussion questions
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