Approximation algorithms for min vector subset-sum over GF(2)
Dumer, Miccancio and Sudan showed that the minimum distance of a linear code is not approximable to any constant factor in randomized polynomial time, unless $\mathsf{NP} = \mathsf{RP}$. The minimum distance problem is the same as your problem if the $v_i$'s form the columns of the parity check matrix and $y = 0$.