Hexacode: Difference between revisions
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In [[coding theory]], the '''hexacode''' is 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 |
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 |
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:<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> |
:<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> |
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It is a 3-dimensional subspace of the vector space of dimension 6 over |
It is a 3-dimensional subspace of the [[vector space]] of dimension 6 over |
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<math>GF(4)</math>. |
<math>GF(4)</math>. |
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Then <math>H</math> contains 45 [[codewords]] of [[weight]] 4, 18 codewords of weight 6 and |
Then <math>H</math> contains 45 [[Code word (communication)|codewords]] of [[Hamming weight|weight]] 4, 18 codewords of weight 6 and |
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the zero word. The full [[automorphism group]] of the hexacode is |
the zero word. The full [[automorphism group]] of the hexacode is |
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<math>3.S_6</math>. The hexacode can be used to describe the [[Miracle Octad Generator]] |
<math>3.S_6</math>. The hexacode can be used to describe the [[Miracle Octad Generator]] |
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==References== |
==References== |
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*{{cite book | first = John H. | last = Conway | authorlink = John Horton Conway | |
*{{cite book | first = John H. | last = Conway | authorlink = John Horton Conway |author2=Sloane, Neil J. A. |authorlink2=Neil Sloane | year = 1998 | title = Sphere Packings, Lattices and Groups | url = https://archive.org/details/spherepackingsla0000conw_b8u0 | url-access = registration | edition = (3rd ed.) | publisher = Springer-Verlag | location = New York | isbn = 0-387-98585-9}} |
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[[Category:Coding theory]] |
[[Category:Coding theory]] |
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Latest revision as of 01:55, 21 November 2023
In coding theory, the hexacode is a length 6 linear code of dimension 3 over the Galois field of 4 elements defined by
It is a 3-dimensional subspace of the vector space of dimension 6 over . Then contains 45 codewords of weight 4, 18 codewords of weight 6 and the zero word. The full automorphism group of the hexacode is . The hexacode can be used to describe the Miracle Octad Generator of R. T. Curtis.
References
[edit]- 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.