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** $L $ in $\mathbb{R}^n $ is the set of all integral linear combinations of n linearly independent vectors $\{ v_1,\ldots,v_n \} $, that is

$L = \mathbb{Z} v_1 \oplus \ldots \oplus \mathbb{Z} v_n $

The theta function of the lattice is the power series

$\Theta_L(q) = \sum_l a_l q^l $

with $a_l $ being the number of vectors in $L $ of squared length $l $. If all squared lengths are even integers, the lattice is called **even** and if it has one point per unit volume, we call it **unimodular**. The theta function of an even unimodular lattice is a modular form. One of the many gems from Conway’s book The sensual (quadratic) form is the chapter “Can You Hear the Shape of a Lattice?” or in other words, whether the theta function determines the lattice.

Ernst Witt knew already that there are just two even unimodular lattices in 16 dimensions : $E_* \oplus E_8 $ and $D_{16}^+ $ and as there is just one modular form of weigth 8 upto scalars, the theta function cannot determine the latice in 16 dimensions. The number of dimensions for a counterexamle was sunsequently reduced to 12 (Kneser), 8 (Kitaoka),6 (Sloane) and finally 4 (Schiemann).

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 $\mathbb{F}_3 = \\{ 0,+,- \\} $

$\begin{matrix} 0~0 0 0 & 0~+ + + & 0~- – – \\\ +~0 + – & +~+ – 0 & +~- 0 + \\\ -~0 – + & -~+ 0 – & -~- + 0 \end{matrix} $

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 $\mathbb{F}_3 $). Now take four mutually orthogonal vectors in $\mathbb{R}^4 $ with square lengths

$e_a.e_a = \frac{1}{12}, e_b.e_b = \frac{7}{12}, e_c.e_c = \frac{13}{12}, e_d.e_d= \frac{19}{12} $

and denote with $~(w,x,y,z) $ the vector $w e_a + x e_b + y e_c + z e_d $. Now consider the two lattices $L^+ $ respectively $L^- $ spanned by the vectors

$~(3,-1,-1,-1),(1,3,1,-1),(1,-1,3,1),(1,1,-1,3) $ resp. $~(-3,-1,-1,-1),(1,-3,1,-1),(1,-1,-3,1),(1,1,-1,-3) $

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 $L^+ $ and $L^- $ given by the rule : **change the sign of the first coordinate that is divisible by 3**. As a direct consequence, the theta functions of these two lattices are equal.

Yet, these lattices cannot be isometric. One verfies that the only vectors of norm 4 in $L^+ $ are $\pm (3,-1,-1,-1) $ and those of norm 8 are $\pm (1,3,1,-1) $ and one computes that their inproduct is

$~(3,-1,-1,-1).(1,3,1,-1)=-1 $

Similarly, the only vectors of norm 4 in $L^- $ are $\pm (-3,-1,-1,-1) $ and those of norm 8 are $\pm (1,-3,1,-1) $ whereas their inproduct is

$~(-3,-1,-1,-1).(1,-3,1,-1) = 2 $

so the two lattices are different!

**Reference**

John H. Conway, “The sensual (quadratic) form” second lecture “Can you hear the shape of a lattice?”

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