# Lexicographic code

Lexicographic codes or lexicodes are greedily generated error-correcting codes with remarkably good properties. They were produced independently by Levenshtein[1] and Conway and Sloane [2] 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. [3]

## Notes

1. ^ V.I. Levenstein. A class of systematic codes. Soviet Math. Dokl, 1(1):368-371, 1960.
2. ^ J.H. Conway and N.J.A Sloane. Lexicographic codes: error-correcting codes from game theory. IEEE Transactions on Information Theory, 32:337-348, 1986.
3. ^ Ari Trachtenberg, Designing Lexicographic Codes with a Given Trellis Complexity, IEEE Transactions on Information Theory, January 2002.