Hexacode

Channel:

WikiReader

Subscribers:

9,560

Published on March 27, 2022 9:24:50 PM ● Video Link: https://www.youtube.com/watch?v=X2g71ICB6oM

Duration: 0:02

2 views

0

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.

Source: https://en.wikipedia.org/wiki/Hexacode
Created with WikipediaReaderReborn (c) WikipediaReader