On the gaps of the spectrum of volumes of trades
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A pair {T_0,T_1} of disjoint collections of k-subsets (blocks) of a set V of cardinality v is called a t-(v,k) trade or simply a t-trade if every t-subset of V is included in the same number of blocks of T_0 and T_1. The cardinality of T_0 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 2^t+1-2^i and less than 2^t+1-2^i-1 and derive restrictions on the t-trade volumes that are less than 2^t+1+2^t-1.
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