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2019-06-09T13:30:11Z

Created page with "'''Lexicographic codes''' or lexicodes are greedily generated error-correcting codes with remarkably good properties. They were produced independently by Levenshtein and C..."

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'''Lexicographic codes''' or lexicodes are greedily generated [[error-correcting code]]s with remarkably good properties. They were produced independently by
Levenshtein and Conway and Sloane and are known to be [[linear code | linear]] over some [[finite field]]s.

== Construction ==
A lexicode of minimum distance ''d'' and length ''n'' over a [[finite field]] is generated by starting with the all-zero vector and iteratively adding the next vector (in [[lexicographic order]]) of minimum [[Hamming distance]] ''d'' from the vectors added so far. As an example, the length-3 lexicode of minimum distance 2 would consist of the vectors marked by an "X" in the following example:

:{| class="wikitable"
|-
! Vector
! In code?
|-
| 000
| X
|-
| 001
|
|-
| 010
|
|-
| 011
| X
|-
| 100
|
|-
| 101
| X
|-
| 110
| X
|-
| 111
|
|}

Since lexicodes are linear, they can also be constructed by means of their [[Basis (linear algebra) | basis]].

== Notes ==
<references/>

== External links ==
*[http://burtleburtle.net/bob/math/lexicode.html Bob Jenkins table of binary lexicodes]
*[http://ipsit.bu.edu/comp.html On-line generator for lexicodes and their variants]
*
*[http://ipsit.bu.edu/phdthesis_html/phdthesis_html.html Error-Correcting Codes on Graphs: Lexicodes], [http://oeis.org/search?q=Trellises Trellises and Factor Graphs]

==Source==

[http://wikipedia.org/ http://wikipedia.org/]
[[Category:Error-correcting codes]]
==See Also on BitcoinWiki==
* [[ALAX]]
* [[IAT]]
* [[FRELDO]]
* [[LEXIT]]
* [[Howdoo]]

Lexicographic code - Revision history

Admin: Created page with "'''Lexicographic codes''' or lexicodes are greedily generated error-correcting codes with remarkably good properties. They were produced independently by Levenshtein and C..."