Linear Codes with Exponentially Many Light Vectors

Let C be a code over q of length n and distance d = d(C): The (Hamming) distance distribution of the code is an (n + 1)-vector (A0 = 1; A1; : : : ; An); where Aw = Aw(C) := (]C) 1jf(x; x0) 2 C : d(x; x0) = wgj: Of course Aw = 0 if 1 w d 1: Let fCng be a family of binary linear codes of growing length n and let dn = d(Cn): G. Kalai and N. Linial [2] conjectured that for any such family the number Adn is subexponential in n, i.e., that for any > 0 there is a number N ( ) such that for all n > N ( ) we have logAdn n (if the base of logarithms is missing, it is 2 throughout). They also made a similar conjecture about unrestricted (i.e., not necessarily linear) codes and wrote \The [asymptotic] distance distribution near the minimum distance remains a great mystery." While we now know a little more about the distance distribution of codes in general [1], [3], this claim is still very much true. The above conjectures, however, are not as will be shown below. Let