Embedding basics
The five-step recipe, determinant and Gaussian heuristic, and a first embedding you can check by hand.
Dimension 3Embedding guides
Build the lattice so the answer is its shortest vector, then let reduction find it.
Lattice reduction is a general-purpose tool: give it a basis and it returns short vectors. The craft is in the embedding, the choice of basis that makes the answer to your problem one of those short vectors. These guides grow from a three-dimensional example to systems with a hundred unknowns. Every example runs with the g6k-rs recipes, and reading the basis and LLL chapters first will help.
Integer relations
Recover the minimal polynomial of an algebraic number from its decimal digits, and choose the scaling factor that makes it work.
Dimension 5–9Simultaneous Diophantine approximation
Find one denominator that approximates several real numbers at once, and weight the coordinates so the answer is shortest.
Dimension 4–12Closest vectors by Kannan embedding
Turn a closest-vector problem into a shortest-vector problem with one extra coordinate, and compare it with Babai decoding.
Dimension 20–40Short solutions of Ax = b
Kernel lattices, homogenization, sub-dimension choice and progressive BKZ with sieving on larger instances.
Dimension 40–100