Hexacode

In <a href="/facts/Coding_theory/SBqStoGL">coding theory</a>, the hexacode is a length 6 <a href="/facts/Linear_code/HoFI3eyF">linear code</a> of dimension 3 over the <a href="/facts/Galois_field/UkMGJo3m">Galois field</a> 
 
 
 
 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 <a href="/facts/Vector_space/M1pxMLx2">vector space</a> of dimension 6 over

G
 F
 (
 4
 )
 
 
 {\displaystyle GF(4)}
 
.
Then 
 
 
 
 H
 
 
 {\displaystyle H}
 
 contains 45 <a href="/facts/Code_word_(communication)/sASREopJ">codewords</a> of <a href="/facts/Hamming_weight/dJIW7qg4">weight</a> 4, 18 codewords of weight 6 and
the zero word. The full <a href="/facts/Automorphism_group/65vImKwe">automorphism group</a> of the hexacode is

3.
 
 A
 
 6
 
 
 
 
 {\displaystyle 3.A_{6}}
 
. The hexacode can be used to describe the <a href="/facts/Miracle_Octad_Generator/9nWzr2fA">Miracle Octad Generator</a>
of R. T. Curtis.

<ul><li><a href="/facts/John_Horton_Conway/g3PA1zoK">Conway, John H.</a>; <a href="/facts/Neil_Sloane/kLWw29Me">Sloane, Neil J. A.</a> (1998). <a href="https://archive.org/details/spherepackingsla0000conw_b8u0">Sphere Packings, Lattices and Groups</a> (3rd ed.). New York: Springer-Verlag. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-387-98585-9.</li></ul>

Hexacode open-in-new

Hexacode