Linear Codes and Character Sums

Let V be an rn-dimensional linear subspace of Zn 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 −Ω( r2 log(1/r)+1 n) . We also answer a question of Ben-Or, and show that if r > 12 , then for every k, at most Cr · |V | √n vectors of V have weight k. Our work draws on a simple connection between extremal properties of linear subspaces of Zn 2 and the distribution of values in short sums of Z n 2 -characters. Hebrew University, Jerusalem 91904, Israel. E-mail: [email protected]. Supported in part by grants from the Israeli Academy of Sciences and the Binational Science Foundation Israel-USA. Institute for Advanced Study, Princeton, NJ 08540. E-mail: [email protected]. This work was done while the author was a student in the Hebrew University of Jerusalem, Israel.