hexacode

hexacode

The

hexacode

is a 3-dimensional

linear code

of length (

http://planetmath.org/LinearCode

) 6, defined over the field

𝔽

4

, all of whose

codewords

have weight 0, 4, or 6. It is uniquely determined by these

properties

, up to

monomial

linear transformations (

http://planetmath.org/MonomialMatrix

) and in

𝔽

4

. The hexacode is crucial to the construction of the

extended binary Golay code

via Curtis’

Miracle Octad Generator

. The exposition below follows (

[

1

]

, Chapter 11). Another for the hexacode is (

[

2

]

, Chapter 4).

1

Construction of the hexacode

There are several constructions of the hexacode, all leading to the same result. In the following, we write the elements of

𝔽

4

as

{

0

,

1

,

ω

,

ω

¯

}

where

ω

is a

cube root

of unity. We write elements of the hexacode as elements of

𝔽

4

6

, separated into three of two.

1. The span of the

elements

ω

⁢

ω

¯

⁢

ω

⁢

ω

¯

⁢

ω

⁢

ω

¯

ω

⁢

ω

¯

⁢

ω

¯

⁢

ω

⁢

ω

¯

⁢

ω

ω

¯

⁢

ω

⁢

ω

⁢

ω

¯

⁢

ω

¯

⁢

ω

ω

¯

⁢

ω

⁢

ω

¯

⁢

ω

⁢

ω

⁢

ω

¯

2. The elements of

𝔽

4

6

of the form

(

a

,

b

,

c

,

ϕ

⁢

(

1

)

,

ϕ

⁢

(

ω

)

,

ϕ

⁢

(

ω

¯

)

)

where

a

,

b

,

c

∈

𝔽

4

and

ϕ

⁢

(

x

)

=

a

⁢

x

2

+

b

⁢

x

+

c

.

3. The elements

a

⁢

b

⁢

c

⁢

d

⁢

e

⁢

f

of

𝔽

4

6

which

satisfy

the three rules

a

+

b

=

c

+

d

=

e

+

f

=

s

a

+

c

+

e

=

a

+

d

+

f

=

b

+

c

+

f

=

b

+

d

+

e

=

ω

⁢

s

b

+

d

+

f

=

b

+

c

+

e

=

a

+

d

+

e

=

a

+

c

+

f

=

ω

¯

⁢

s

Here

s

is called the

of the codeword.

An element of the hexacode is called a

hexacodeword

.

2

Justification of a hexacodeword

It is not difficult to show that all of the above constructions give the same 3-dimensional linear code of 6. However, it is somewhat tedious to use one of above constructions to determine whether a given element of

𝔽

4

6

is in the hexacode. Instead, it is possible to show that an element of

𝔽

4

6

is a hexacodeword if and only if it satisfies the

shape

and

sign

rules below. Using these rules, one can (with some practice) quickly distinguish hexacodewords from non-hexacodewords.

The shape rule says that, up to

permutations

of the three blocks and flips of the elements within a block, every hexacodeword has one of the shapes

00  00  00

00

⁢

a

⁢

a

⁢

a

⁢

a

0

⁢

a

⁢

0

⁢

a

⁢

b

⁢

c

b

⁢

c

⁢

b

⁢

c

⁢

b

⁢

c

a

⁢

a

⁢

b

⁢

b

⁢

c

⁢

c

where

a

,

b

,

c

are

1

,

ω

,

ω

¯

in some .

The sign rule says that in every hexacodeword, either:

•

all 3 blocks have sign 0, or

•

the

product

of the signs of the three blocks is

positive

.

The sign of a block is determined as follows:

•

+ for

0

⁢

a

or

a

⁢

b

where

b

=

a

⁢

ω

•

- for

a

⁢

0

or

a

⁢

b

where

b

=

a

⁢

ω

¯

•

0 for

00

or

a

⁢

a

where

a

≠

0

.

For example,

•

00  11

⁢

ω

⁢

ω

and

01  0

⁢

ω

⁢

1

⁢

ω

¯

are not hexacodewords because they fail the shape rule.

•

01  01

⁢

ω

⁢

ω

¯

satisfies the shape rule, and the signs are + + +, so it is a hexacodeword.

•

1

⁢

ω

⁢

ω

⁢

1  1

⁢

ω

satisfies the shape rule, and the signs are + - +, so it is not a hexacodeword.

•

00  11  11

satisfies the shape rule, and the signs are 0 0 0, so it is a hexacodeword.

•

ω

⁢

0

⁢

ω

⁢

0

⁢

ω

¯

⁢

1

satisfies the shape rule, and the signs are - - +, so it is a hexacodeword.

3

Completion of a partial hexacodeword

The hexacode has the property that the following two problems always have a unique

solution

.

1. (3-problem) Given values in any 3 of the 6 positions,

complete

it to a full hexacodeword.

2. (5-problem) Given values in any 5 of the 6 positions, complete it to a full hexacodeword

after possibly changing one of the given values

.

Conway says that the best method for solving these is to simply ”guess the correct answer, then justify it” (using the shape and sign rules), though he also does give systematic algorithms for solving them.

Examples of 3-problems:

01  1

⁢

?

⁢

?

⁢

?

→

01  10

⁢

ω

¯

⁢

ω

?

⁢

?

⁢

1

⁢

ω

¯

⁢

?

⁢

0

→

0

⁢

ω

⁢

1

⁢

ω

¯

⁢

ω

⁢

0

1

⁢

?

⁢

ω

⁢

ω

⁢

?

⁢

?

→

11

⁢

ω

⁢

ω

⁢

ω

¯

⁢

ω

¯

Examples of 5-problems:

00  11

⁢

ω

⁢

?

→

00  11  11

⁢

(position 5 changed)

0

⁢

?

⁢

1

⁢

ω

¯

⁢

ω

⁢

ω

¯

→

0

⁢

ω

⁢

1

⁢

ω

¯

⁢

ω

⁢

0

⁢

(position 6 changed)

1

⁢

ω

⁢

?

⁢

1  1

⁢

ω

→

1

⁢

ω

⁢

ω

¯

⁢

1  1

⁢

ω

⁢

(no position changed)

References

1

J. H. Conway and N. J. A. Sloane.

Sphere Packings

, Lattices, and Groups. Springer-Verlag, 1999.

2

Robert L. Griess, Jr. Twelve

Sporadic Groups

. Springer-Verlag, 1998.

Title

hexacode

Canonical

name

Hexacode

Date of creation

2013-03-22 18:43:08

Last modified on

2013-03-22 18:43:08

Owner

monster (22721)

Last modified by

monster (22721)

Numerical id

15

Author

monster (22721)

Entry type

Definition

Classification

msc 94B05

Related topic

MiracleOctadGenerator

Related topic

BinaryGolayCode

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