They were produced independently by Vladimir Levenshtein[1] and by John Horton Conway and Neil Sloane.
[2] A lexicode of length n and minimum distance d 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: Here is a table of all n-bit lexicode by d-bit minimal hamming distance, resulting of maximum 2m codewords dictionnary.
All odd d-bit lexicode distances are exact copies of the even d+1 bit distances minus the last dimension, so an odd-dimensional space can never create something new or more interesting than the d+1 even-dimensional space above.
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.