Lexicographic code


Lexicographic codes or lexicodes are greedily generated error-correcting codes with remarkably good properties. They were produced independently by
Vladimir Levenshtein and by John Horton Conway and Neil Sloane. The binary lexicographic codes are linear codes, and include the Hamming codes and the binary Golay codes.

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 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:
Since lexicodes are linear, they can also be constructed by means of their basis.

Combinatorial game theory

The theory of lexicographic codes is closely connected to combinatorial game theory. In particular, the codewords in a binary lexicographic code of distance d encode the winning positions in a variant of Grundy's game, played on a collection of heaps of stones, in which each move consists of replacing any one heap by at most d − 1 smaller heaps, and the goal is to take the last stone.