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connection to combinatorial game theory |
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Here is a table of n-bit lexicode by d-bit minimal hamming distance, resulting of maximum 2<sup>m</sup> codewords dictionnary.
For example, extended Hamming code (n=8,d=4,m=4) and especially Golay code (n=24,d=8,m=12) shows exceptional compactness compared to neighbors.
:{| class="wikitable"
|-
! n \ d
! 1
! 2
! 3
! 4
! 5
! 6
! 7
! 8
! 9
! 10
! 11
! 12
! 13
! 14
! 15
! 16
! 17
! 18
|-
! 1
| 1
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|-
! 2
| 2
| 1
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|-
! 3
| 3
| 2
| 1
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|-
! 4
| 4
| 3
| 1
| 1
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|-
! 5
| 5
| 4
| 2
| 1
| 1
|
|
|
|
|
|
|
|
|
|
|
|
|
|-
! 6
| 6
| 5
| 3
| 2
| 1
| 1
|
|
|
|
|
|
|
|
|
|
|
|
|-
! 7
| 7
| 6
| 4
| 3
| 1
| 1
| 1
|
|
|
|
|
|
|
|
|
|
|
|-
! 8
| 8
| 7
| 4
| {{good|4}}
| 2
| 1
| 1
| 1
|
|
|
|
|
|
|
|
|
|
|-
! 9
| 9
| 8
| 5
| 4
| 2
| 2
| 1
| 1
| 1
|
|
|
|
|
|
|
|
|
|-
! 10
| 10
| 9
| 6
| 5
| 3
| 2
| 1
| 1
| 1
| 1
|
|
|
|
|
|
|
|
|-
! 11
| 11
| 10
| 7
| 6
| 4
| 3
| 2
| 1
| 1
| 1
| 1
|
|
|
|
|
|
|
|-
! 12
| 12
| 11
| 8
| 7
| 4
| 4
| 2
| 2
| 1
| 1
| 1
| 1
|
|
|
|
|
|
|-
! 13
| 13
| 12
| 9
| 8
| 5
| 4
| 3
| 2
| 1
| 1
| 1
| 1
| 1
|
|
|
|
|
|-
! 14
| 14
| 13
| 10
| 9
| 6
| 5
| 4
| 3
| 2
| 1
| 1
| 1
| 1
| 1
|
|
|
|
|-
! 15
| 15
| 14
| 11
| 10
| 7
| 6
| 5
| 4
| 2
| 2
| 1
| 1
| 1
| 1
| 1
|
|
|
|-
! 16
| 16
| 15
| 11
| 11
| 8
| 7
| 5
| 5
| 2
| 2
| 1
| 1
| 1
| 1
| 1
| 1
|
|
|-
! 17
| 17
| 16
| 12
| 11
| 9
| 8
| 6
| 5
| 3
| 2
| 2
| 1
| 1
| 1
| 1
| 1
| 1
|
|-
! 18
| 18
| 17
| 13
| 12
| 9
| 9
| 7
| 6
| 3
| 3
| 2
| 2
| 1
| 1
| 1
| 1
| 1
| 1
|-
! 19
| 19
| 18
| 14
| 13
| 10
| 9
| 8
| 7
| 4
| 3
| 2
| 2
| 1
| 1
| 1
| 1
| 1
| 1
|-
! 20
| 20
| 19
| 15
| 14
| 11
| 10
| 9
| 8
| 5
| 4
| 3
| 2
| 2
| 1
| 1
| 1
| 1
| 1
|-
! 21
| 21
| 20
| 16
| 15
| 12
| 11
| 10
| 9
| 5
| 5
| 3
| 3
| 2
| 2
| 1
| 1
| 1
| 1
|-
! 22
| 22
| 21
| 17
| 16
| 12
| 12
| 11
| 10
| 6
| 5
| 4
| 3
| 2
| 2
| 1
| 1
| 1
| 1
|-
! 23
| 23
| 22
| 18
| 17
| 13
| 12
| 12
| 11
| 6
| 6
| 5
| 4
| 2
| 2
| 2
| 1
| 1
| 1
|-
! 24
| 24
| 23
| 19
| 18
| 14
| 13
| 12
|{{won|12}}
| 7
| 6
| 5
| 5
| 3
| 2
| 2
| 2
| 1
| 1
|-
! 25
| 25
| 24
| 20
| 19
| 15
| 14
| 12
| 12
| 8
| 7
| 6
| 5
| 3
| 3
| 2
| 2
| 1
| 1
|-
! 26
| 26
| 25
| 21
| 20
| 16
| 15
| 12
| 12
| 9
| 8
| 7
| 6
| 4
| 3
| 2
| 2
| 2
| 1
|-
! 27
| 27
| 26
| 22
| 21
| 17
| 16
| 13
| 12
| 9
| 9
| 7
| 7
| 5
| 4
| 3
| 2
| 2
| 2
|-
! 28
| 28
| 27
| 23
| 22
| 18
| 17
| 13
| 13
| 10
| 9
| 8
| 7
| 5
| 5
| 3
| 3
| 2
| 2
|-
! 29
| 29
| 28
| 24
| 23
| 19
| 18
| 14
| 13
| 11
| 10
| 8
| 8
| 6
| 5
| 4
| 3
| 2
| 2
|-
! 30
| 30
| 29
| 25
| 24
| 19
| 19
| 15
| 14
| 12
| 11
| 9
| 8
| 6
| 6
| 5
| 4
| 2
| 2
|-
! 31
| 31
| 30
| 26
| 25
| 20
| 19
| 16
| 15
| 12
| 12
| 10
| 9
| 6
| 6
| 6
| 5
| 3
| 2
|-
! 32
| 32
| 31
| 26
| 26
| 21
| 20
| 16
| 16
| 13
| 12
| 11
| 10
| 7
| 6
| 6
| 6
| 3
| 3
|-
! 33
| ...
| 32
| ...
| 26
| ...
| 21
| ...
| 16
| ...
| 13
| ...
| 11
| ...
| 7
| ...
| 6
| ...
| 3
|}
All odd d-bit distance lexicodes are exact copies of the even d+1 bit distance minus the last dimension, so
an odd-dimensional space can never create something new or more interesting than the d+1 even-dimensional space.
Since lexicodes are linear, they can also be constructed by means of their [[Basis (linear algebra) | basis]].<ref>{{citation
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