Linear Codes and Character Sums

Let V be an rn-dimensional linear subspace of $Z^{n}_{2}$. 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 $2^{{ - \Omega {\left( {\frac{{r^{2} }} {{\log {\left( {1/r} \right)} + 1}}n} \right)}}}$.

We also answer a question of Ben-Or and show that if $r > \frac{1} {2}$, then for every k, at most $C_{r} \cdot \frac{{{\left| V \right|}}} {{{\sqrt n }}}$ vectors of V have weight k.

Our work draws on a simple connection between extremal properties of linear subspaces of $Z^{n}_{2}$ and the distribution of values in short sums of $Z^{n}_{2}$-characters.