# Lattice

# 2.1 Z_n

Z_n is the subset of R_n with all integer coordinates.

It is given the closest lattice point v ∈ Z_n to arbitrary point u ∈ R_n.

Let d ∈ R_n, d = u - v.

Find k = argmax |d_k|.

Let e ∈ Z_n, with e_k = 1, d_k > 0; e_k = -1, d_k < 0; e_i = 0, i ≠ k.

Then w ∈ Z_n, w = v + e is the second-closest lattice point to u.

# 2.2 D_n

D_n is the subset of Z_n with sum of coordinates even.

It is given the closest lattice point v ∈ D_n to arbitrary point u ∈ R_n.

Let d ∈ R_n, d = u - v.

Find k = argmax |d_k|. Find j = argmax |d_j, j ≠ k|.

Let e ∈ D_n, with e_k = 1, d_k > 0; e_k = -1, d_k < 0; e_j = 1, d_j > 0; e_j = -1, d_j < 0; e_i = 0, i ≠ j and i ≠ k.

Then w ∈ D_n, w = v + e is the second-closest lattice point to u.

# 2.3 A_n

A_n is the subset of Z_n+1 with sum of coordinates 0.

Let R’_n be the subset of R_n+1 with sum of coordinates 0.

It is given the closest lattice point v ∈ A_n to arbitrary point u ∈ R’_n.

Let d ∈ R’_n, d = u - v.

Find k = argmax d_k. Find j = argmin d_k.

Let e ∈ Z_n+1, e_k = 1; e_i = 0, i ≠ k. Let f ∈ Z_n+1, f_j = -1; f_i = 0, i ≠ j.

Then w ∈ A_n, w = v + e + f is the second-closest lattice point to u.

# 2.4 E₆

Future work.

# 2.5 E₇

E₇ can be expressed as the union of 8 copies of 2Z₇.

It is given the closest lattice point v ∈ E₇ to arbitrary point u ∈ R₇.

Let w’ ∈ E₇ be the second-closest lattice point in the same coset of 2Z₇.

Let w’’ ∈ E₇ be the closest lattice point in all the other cosets of 2Z₇.

Then w ∈ E₇, the closest to u of w’ and w’’, is the second-closest lattice point to u.

# 2.6 E₈

E₈ can be expressed as the union of 2 copies of D₈.

It is given the closest lattice point v ∈ E₈ to arbitrary point u ∈ R₈.

Let w’ ∈ E₈ be the second-closest lattice point in the same coset of D₈.

Let w’’ ∈ E₈ be the closest lattice point in the other coset of D₈.

Then w ∈ E₈, the closest to u of w’ and w’’, is the second-closest lattice point to u.