Linear Codes with Exponentially Many Light Vectors

G. Kalai and N. Linial (1995) put forward the following conjecture: Let {C sub n } be a sequence of binary linear codes of distance d sub n and A sub d sub n be the number of vectors of weight d sub n in C sub n , then log sub 2 A sub d sub n = l(n). We disprove this by constructing a family of linear codes from geometric Goppa codes in which the number of vectors of minimum weight grows exponentially with the length.