Linear codes and character sums

Let V be an rn-dimensional linear subspace of Z₂ⁿ. Suppose the smallest Hamming weight of non-zero vectors in V is d. (In coding-theoretic terminology, V is a linear code of length n, rate r and distance d.) We settle two extremal problems on such spaces. First, we prove a (weak form) of a conjecture by Kalai and Linial and show that the fraction of vectors in V with weight d is exponentially small. Specifically, in the interesting case of a small r, this fraction does not exceed 2^{-Ω(r2/log(1/r)+1n)}. We also answer a question of Ben-Or and show that if r > 1/2, then for every k, at most C_r · |V|/√n vectors of V have weight k. Our work draws on a simple connection between extremal properties of linear subspaces of Z₂ⁿ and the distribution of values in short sums of Z₂ⁿ-characters.