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^[1] and Conway and Sloane ^[2] and are known to be linear over some finite fields.

[edit] Construction

A lexicode of minimum distance d and length $n$ over a finite field $\mathbb{F}$ 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]

[edit] Notes

^ V.I. Levenstein. A class of systematic codes. Soviet Math. Dokl, 1(1):368-371, 1960.
^ 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.
^ A. Trachtenberg, Designing Lexicographic Codes with a Given Trellis Complexity, IEEE Transactions on Information Theory, January 2002.