The cube says no
Three impossibility theorems on Q6
Walk the three acts: the maximum code of 8 words, the bipartition every edge crosses, and Pólya's 14 necklaces.
Not everything one wishes for exists. The first impossibility theorems of the site, in three verified acts: how many hexagrams can correct each other, why there are no odd cycles of mutations, and how many distinct necklaces the six lines make when rotated.
The sphere-packing bound says that at distance 3 there is room for at most 64 / (1 + 6) = 9.14, that is 9 words. But 7 does not divide 64: there is no perfect code. The real maximum is 8, proved by exhaustive search. These 8 hexagrams correct each other: any single-line error is detected and repaired.
tap any hexagram: if it is a single-line error, it is repaired to the nearest codeword
The sphere-packing bound gives at most 9 words at distance 3 in Q6 (7 does not divide 64: there is no perfect code); the real maximum is 8, proved by exhaustive search, and a code of 8 corrects any single-line error.
verificar_cubo_noThe 32 hexagrams with an even number of yang lines and the 32 with an odd number form a bipartition: the 192 edges all cross from one half to the other, so Q6 is bipartite and there are no odd cycles of mutations.
verificar_cubo_noUnder rotation of the 6 lines there are exactly 14 necklaces (Pólya's formula = direct enumeration) and 13 bracelets once the reflection is added.
verificar_cubo_noThe types of claim and the full bibliography (APA) are in Foundations.