Weight distribution and dual distance

Several results in coding theory (e.g. the Carlitz-Uchiyama bound) show that the weight distributions of certain algebraic codes of lengthn are concentrated aroundn/2 within a range of width n. It is proved in this article that the extreme weights of a linear binary code of sufficiently high dual distance cannot be too close ton/2, the gap being of order n. The tools used involve the Pless identities and the orthogonality properties of Krawtchouk polynomials, as well as estimates on their zeroes. As a by-product upper bounds on the minimum distance of self-dual binary codes are derived.