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Abstract  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 than2   t  +  1    −   2  iand less than2   t  +  1    −   2   i  −  1and derive restrictions on the  t ‐trade volumes that are less than2   t  +  1    +   2   t  −  1.

On the gaps of the spectrum of volumes of trades