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Lexicographic code
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Lexicographic codes or lexicodes are greedily generated error-correcting codes with remarkably good properties. They were produced independently by Levenshtein and Conway and Sloane and are known to be linear over some finite fields.
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:
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.
Notes
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External links
- Bob Jenkins table of binary lexicodes
- On-line generator for lexicodes and their variants
- Error-Correcting Codes on Graphs: Lexicodes, Trellises and Factor Graphs
