Error-correcting codes can be used to construct interesting lattices, the best known example being the Leech lattice constructed from the binary Golay code. Recall that a lattice
The theta function of the lattice is the power series
with
Ernst Witt knew already that there are just two even unimodular lattices in 16 dimensions :
Sloane and Conway found an elegant counterexample in dimension 4 using two old friends : the tetracode and the taxicab number 1729 = 7 x 13 x 19.
Recall that the tetracode is a one-error correcting code consisting of the following nine words of length four over
The first element (which is slightly offset from the rest) is the slope s of the words, and the other three digits cyclically increase by s (in the field
and denote with
then it follows that if we reduce any vector in either lattice modulo 3 we get a tetracode word. Using this fact it is not too difficult to show that there is a length preserving bijection between
Yet, these lattices cannot be isometric. One verfies that the only vectors of norm 4 in
Similarly, the only vectors of norm 4 in
so the two lattices are different!
Reference
John H. Conway, โThe sensual (quadratic) formโ second lecture โCan you hear the shape of a lattice?โ
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