Linear Codes and Character Sums

• Authors: Nathan Linial;Alex Samorodnitsky
• Venue: Combinatorica
• Year: 2002

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.