Lexicographic codes: Error-correcting codes from game theory

Lexicographic codes, or lexicodes, are defined by various versions of the greedy algorithm. The theory of these codes is closely related to the theory of certain impartial games, which leads to a number of surprising properties. For example, lexicodes over an alphabet of size
B=2^{a}
are closed under addition, while if
B = 2^{2^{a}}
the lexicodes are closed under multiplication by scalars, where addition and multiplication are in the nim sense explained in the text. Hamming codes and the binary Golay codes are lexieodes. Remarkably simple constructions are given for the Steiner systems
S(5,6,12)
and
S(5,8,24)
. Several record-breaking constant weight codes are also constructed.