# What is the meaning for a vector mod a matrix in a lattice?

I'm reading about the lattice recently.In the paper, it gives a method of a vector mod a matrix:

⃗c mod B as ⃗c−⌊⃗c×B^(−1)⌉×B = [⃗c×B^(−1)]×B.

I know that a integer A mod the other integer B is A+-kB, to make A in the defined field B.

Can anyone tell me what is the meaning of a vector mod a matrix and the mid term ⌊⃗c×B^(−1)⌉ means?:(

Tks.

A vector u mod a matrix B is ​ u ± Bv . ​ ​ ​ We are modding an element of the vector space's
additive group by the range of the restriction of B to vectors whose entries are all integers.
The parallelepiped given by B's columns is a fundamental domain for the range of that restriction.

⌊⃗c×B−1⌉ is probably rounding each entry of c×B−1 to the nearest integer. ​ If that's correct, then they are using the lattice representation, which corresponds to using [the translation of the parallelepiped I mentioned which sends its center to the origin] as the fundamental domain.

• well, i mean, we always mod the two same thing (like integer to integer), to get the answer. We use a vector mod a matrix to make the vector in the defined range, but now the range is a "matrix". i couldn't image how a matrix to be a "range", is it like a projection in a vector space or another thing? – lafara94130 Jul 15 '15 at 8:38