AbstractA pair of disjoint collections of k‐subsets (blocks) of a set V of cardinality v is called a t‐ trade or simply a t‐trade if every t‐subset of V is included in the same number of blocks of T0 and T1. The cardinality of T0 is called the volume of the trade. Using the weight distribution of the Reed–Muller code, we prove the conjecture that for every i from 2 to t, there are no t‐trades of volume greater than and less than and derive restrictions on the t‐trade volumes that are less than .